Home Mathematics Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
Article Open Access

Nontrivial solutions for a generalized poly-Laplacian system on finite graphs

  • and EMAIL logo
Published/Copyright: July 31, 2025
Download Article (PDF)
Become an author with De Gruyter Brill
Author Information Explore this Subject
Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

We investigate the existence and multiplicity of solutions for a class of the generalized coupled system involving poly-Laplacian and the parameter λ on finite graphs. By using the Mountain pass lemma together with the cut-off technique, we obtain that system has at least a nontrivial weak solution ( u λ , v λ ) for every large parameter λ when the nonlinear term F ( x , u , v ) satisfies superlinear growth conditions only in a neighborhood of origin point (0, 0). We also obtain a concrete form for the lower bound of λ and the trend of ( u λ , v λ ) with the change of λ . Moreover, by using a revised Clark’s theorem together with cut-off technique, we obtain that system has a sequence of solutions tending to 0 for every λ > 0 when the nonlinear term F ( x , u , v ) satisfies sublinear growth conditions only in a neighborhood of origin point (0, 0).

Keywords: coupled system on finite graphs; poly-Laplacian; local nonlinearity near origin; Mountain pass lemma; Clark’s theorem; existence; infinitely many nontrivial solutions
MSC 2010: 35J50; 35J62; 35B38; 35R02

1 Introduction and main results

The existence and multiplicity of solutions for the quasilinear elliptic system with the following form

(1.1) − Δ p u + h 1 ( x ) ∣ u ∣ p − 2 u = λ F u ( x , u , v ) , x ∈ Ω , − Δ q v + h 2 ( x ) ∣ v ∣ q − 2 v = λ F v ( x , u , v ) , x ∈ Ω

have been studied extensively over the past several decades (see [1–4] and the references therein), where Ω ⊂ R N is a domain, Δ p is the so-called p -Laplacian operator, i.e., Δ p = div ( ∣ ∇ u ∣ p − 2 ∇ u ) , 1 < p , q < N , λ > 0 is a parameter, and h i ( x ) ( i = 1 , 2 ) and F are real valued functions satisfying some suitable assumptions. Specially, in [1], Boccardo and Figueiredo investigated the system (1.1) with λ = 1 and h i ( x ) = 0 , i = 1 , 2, where Ω is a bounded domain in R N . They assumed that F : Ω ¯ × R × R → R is of class C 1 and satisfies

∣ F ( x , u , v ) ∣ ≤ C ( 1 + ∣ u ∣ r + ∣ v ∣ s ) ,

where C > 0 is a constant, r ≤ p * , s ≤ q * , p * = p N N − p , and q * = q N N − q . Moreover, there exist constants R > 0 , θ p , and θ q with 1 p * < θ p < 1 p and 1 q * < θ q < 1 q such that

0 < F ( x , u , v ) ≤ θ p u F u ( x , u , v ) + θ q v F v ( x , u , v )

for all x ∈ Ω ¯ and ∣ u ∣ , ∣ v ∣ ≥ R , which is usually called as the Ambrosseti-Rabinowitz condition and

∣ F u ( x , u , v ) ∣ ≤ C 1 + ∣ u ∣ p * − 1 + ∣ v ∣ q * ( p * − 1 ) p * , ∣ F v ( x , u , v ) ∣ ≤ C 1 + ∣ v ∣ q * − 1 + ∣ u ∣ p * ( q * − 1 ) q * .

Under the aforementioned conditions, they established the existence result of a nontrivial solution for system (1.1) by using the Mountain pass lemma. For more results about quasilinear elliptic problems, we refer readers to [5–7] and the references therein.

Corresponding to system (1.1), the following well-known p -Laplacian equation

(1.2) − Δ p u + h ( x ) ∣ u ∣ p − 2 u = λ f ( x , u ) , x ∈ Ω

has also attracted great attention as a result of its important applications to many fields, such as non-Newtonian fluids, reaction–diffusion problems, elastic mechanics, flow through porous media, thermal radiation, and so on. For details, one can see [8] and the references therein. Specially, one of the applications of equation (1.2) is to describe the steady-state flow behavior of fluid with flow density u in the presence of interference, where h ( x ) ∣ u ∣ p − 2 u and λ f ( x , u ) represent two interference sources, respectively. After the interference source h ( x ) ∣ u ∣ p − 2 u disappears, (1.2) reduces to the following p -Laplacian elliptic equation

(1.3) − Δ p u = λ f ( x , u ) , x ∈ Ω ,

which associated with non-Newtonian seepage phenomena [9]. At present, there are many achievements in the study of the existence of solutions to (1.2) and (1.3) (see [10–13] and the references therein). In most of these articles, nonlinear term is required to have growth both near 0 and the infinity about u . In [13], Costa and Wang make some assumptions on the nonlinear term referred only to its behavior in a vicinity of u = 0 . By the cut-off technique and some estimates for solutions as the form ‖ u ‖ ∞ ≤ c 1 λ − c 2 with c i > 0 ( i = 1 , 2 ) , they obtained the number of signed and sign-changing solutions for the elliptic problems (1.3) with p = 2 and f ( x , u ) = f ( u ) under the Dirichlet boundary condition u ( x ) = 0 , x ∈ ∂ Ω , where Ω ⊂ R N is bounded domain. The idea in [13] has been applied to a lot of different problems, for example, Schrödinger equation [14,15] quasilinear elliptic problems involving p -Laplacian operator in R N [16] and fractional-order Kirchhoff-type system [17]. Especially, in [17], Kang et al. applied the idea to the fractional order Kirchhoff-type system with local superlinear nonlinearity. By using the Mountain pass lemma, they established the existence result of a nontrivial weak solution for every given sufficiently large λ . More importantly, they obtained a concrete value of the lower bound of the parameter λ and two estimates of weak solutions.

Recently, the subjects about the partial differential equations on graphs have attracted considerable attentions because of their applications to many fields including data analysis, image processing, neural network, transportation, and so on (see [18–23] and the references therein), and there have been some works, for instance, the existence and nonexistence of solutions of heat equations [24–26], the blow-up and uniqueness properties for solutions of heat equations [27,28], and the existence of solutions of Kazdan-Warner equations [29,30]. Especially, the existence and multiplicity of solutions for the poly-Laplacian system with the following form

(1.4) £ m 1 , p 1 u + h 1 ( x ) ∣ u ∣ p 1 − 2 u = λ F u ( x , u , v ) , x ∈ V , £ m 2 , p 2 v + h 2 ( x ) ∣ v ∣ p 2 − 2 v = λ F v ( x , u , v ) , x ∈ V ,

have also been investigated extensively (see [31–33] and the references therein), where G = ( V , E ) is a finite graph, £ m i , p i are the poly-Laplacian operators defined in section 2, m i ≥ 1 and p i > 1 , h i ( i = 1 , 2 ) , and F are real-valued functions satisfying some assumptions. In [31], when F satisfies the super- ( p , q ) -linear growth condition, by applying the Mountain pass lemma and the Symmetric mountain pass lemma, the authors proved that the system (1.4) has at least one nontrivial solutions and dim W nontrivial solutions, where W is the working space of system (1.4), respectively. In [32], when F satisfies the asymptotically- p -linear growth conditions at infinity, the authors obtained some sufficient conditions about the existence of a nontrivial solution for system (1.4) via the Mountain pass lemma. In [33], by using the Mountain pass lemma and Ekeland’s variational principle, the authors showed that the existence of two nontrivial solutions for system (1.4) involving concave–convex nonlinearity.

In system (1.4), if we set m 1 = m 2 , p 1 = p 2 , h 1 = h 2 , λ = 1 , and u = v , it reduces to the following elliptic equation:

(1.5) £ m , p u + h ( x ) ∣ u ∣ p − 2 u = f ( x , u ) ,

there are also many articles studying problem (1.5) (see [34–37] and the references therein). In [34], Grigor’yan et al. studied the existence of nontrivial solution for problem (1.5) on finite graph by variational method. They assumed that f ( x , t ) satisfies the following conditions:

(f1) There exists some q > p > 1 and M > 0 such that for all x ∈ V and ∣ t ∣ ≥ M ,

0 < q F ( x , t ) ≤ t f ( x , t ) ;

(f2) limsup t → 0 ∣ f ( x , t ) ∣ ∣ t ∣ p − 1 < λ m p , where λ m p is the first eigenvalue of − Δ p defined as follows:

λ m p = inf u ≢ 0 ∫ V ( ∣ ∇ m u ∣ p + h ∣ u ∣ p ) d μ ∫ V ∣ u ∣ p d μ .

They proved that system (1.5) admits a nontrivial solution by the Mountain pass lemma. In [35], by the same method, they established the existence result of strictly positive solutions for nonlinear equations (1.5) with m = 1 and p = 2 . They assumed that h ( x ) satisfies some reasonable assumptions and nonlinear term f ( x , t ) satisfies the following conditions:

(h1) f ( x , 0 ) = 0 , f ( x , t ) is continuous in t , and for any fixed S > 0 , there exist a constant M S such that max t ∈ [ 0 , S ] f ( x , t ) ≤ M S for all x ∈ V ;

(h2) There exists a constant μ > 2 such that for all x ∈ V and t ∈ ( 0 , + ∞ ) ,

0 < μ F ( x , t ) = μ ∫ 0 t f ( x , s ) d s ≤ t f ( x , t ) ;

(h3) limsup t → 0 + 2 F ( x , t ) ∣ t ∣ 2 < λ 1 , where λ 1 is the first eigenvalue of − Δ defined as follows:

λ 1 = inf ∫ V u 2 d μ = 1 ∫ V ( ∣ ∇ u ∣ 2 + h ∣ u ∣ 2 ) d μ .

Moreover, in [36], by using the Nehari method, on a locally finite graph G = ( V , E ) , Zhang and Zhao investigated the convergence of ground state solutions for equation (1.5) with m = 1 , p = 2 , h ( x ) = ς a ( x ) + 1 , and f ( x , u ) = ∣ u ∣ α − 2 u , where ς is a parameter, α ≥ 2 and a ( x ) : V → [ 0 , + ∞ ) . Subsequently, in [37], Han and Shao further studied the nonlinear p -Laplacian equations (1.5) with m = 1 , p ≥ 2 and h ( x ) = ς a ( x ) + 1 on a locally finite graph. Under some conditions like (h1)-(h3) on f and appropriate assumptions on a ( x ) , by the Mountain pass lemma, they proved that the aforementioned equation has a positive solution, and then by the method of Nehari manifold, they also obtained the existence of ground state solution.

Based on the works in [13,17,31,32], the motivation of our work is to consider whether the ideas in [13] can be applied to the generalized poly-Laplacian system (1.4), which is defined on finite graphs. For the super-linear case, by means of the Mountain pass lemma, we show that system (1.4) has at least a nontrivial weak solution for all parameter λ sufficiently large. Moreover, motivated by [17], we manage to find out the specific lower bound of parameter and the tendency of solution with the change of parameter. For the sublinear case, by using a revised Clark’s theorem, we obtain that system (1.4) has a sequence of solutions tending to 0 for every λ > 0 .

Our results develop those results in [17,31,32] from the following aspects:

  1. Our assumptions are different from those in [31,32]. In [31,32], the nonlinear term F needs to have growth both near (0, 0) and infinity, but all of our assumptions on F are just near the origin. It is worthwhile to mention that the existence of infinitely many solutions is not considered in [31,32].

  2. Our results are different from those in [17]. First, we focus on the generalized poly-Laplacian system (1.4) on the finite graphs, which essentially is a discrete structure, instead of the fractional order Kirchhoff-type equation on a bounded interval [ 0 , T ] studied in [17], which is a continuous structure. Moreover, in [17], the authors only considered the case that the nonlinear term F is locally superlinear. But we consider two cases: (I) the nonlinear term F is locally superlinear; (II) The nonlinear term F is locally sublinear. Finally, because of the coupling relationship of u and v and the difference between the definitions of integrals on the graph and in Euclidean space, although there are some similarities with [17] in terms of ideas, proofs in the present article are different from those in [17]. Especially, such difference can be embodied in two ways: (i) establishment of the relationship between ‖ u ‖ ∞ , ‖ v ‖ ∞ and λ (Lemma 3.5); (ii) the proofs of ( C ) c -condition holds for functional I ε (Lemma 3.4). Interestingly, it is necessary to take into account the structure of the graph when one calculates the specific lower bound of parameter, which is different from that in the Euclidean setting (Remark 5.2).

In details, we obtain the following results.

(I) The super-linear case:

Theorem 1.1

Let G = ( V , E ) be a finite graph, m i ≥ 1 and p i > 1 , i = 1 , 2, λ is a parameter with λ > 0 , h i ( x ) ∈ C ( V , R + ) , i = 1 , 2, F : V × R × R → R satisfies the following conditions:

  1. There exists a constant δ > 0 such that F ( x , t , s ) is continuously differentiable in ( t , s ) ∈ R × R with ∣ ( t , s ) ∣ ≤ δ for all x ∈ V ;

  2. There exist q i > p i and M i > 0 , i = 1 , 2 such that

    F ( x , t , s ) ≥ M 1 ∣ t ∣ q 1 + M 2 ∣ s ∣ q 2

    for all x ∈ V and ∣ ( t , s ) ∣ ≤ δ ;

  3. There exist k i ∈ ( p i , q i ) ( i = 1 , 2 ) satisfy min { k 1 , k 2 } > max { p 1 , p 2 } and M j > 0 ( j = 3 , 4 ) such that

    ∣ F t ( x , t , s ) ∣ ≤ M 3 ∣ t ∣ k 1 − 1 + M 4 ∣ s ∣ k 2 ( k 1 − 1 ) k 1 , ∣ F s ( x , t , s ) ∣ ≤ M 3 ∣ t ∣ k 1 ( k 2 − 1 ) k 2 + M 4 ∣ s ∣ k 2 − 1

    for all x ∈ V and ∣ ( t , s ) ∣ ≤ δ , where F t ( x , t , s ) = ∂ F ( x , t , s ) ∂ t and F s ( x , t , s ) = ∂ F ( x , t , s ) ∂ s ;

  4. There exist constants β i > p i , i = 1 , 2 such that

    0 < F ( x , t , s ) ≤ 1 β 1 t F t ( x , t , s ) + 1 β 2 s F s ( x , t , s )

    for all x ∈ V and ∣ ( t , s ) ∣ ≤ δ with ( t , s ) ≠ ( 0 , 0 ) .

Then there exists at least a nontrivial weak solution ( u λ , v λ ) to system (1.4)for all λ > λ * ≔ max { Λ 1 , Λ 2 , Λ 3 , Λ 4 , Λ 5 } and lim λ → ∞ ‖ ( u λ , v λ ) ‖ = 0 = lim λ → ∞ ‖ ( u λ , v λ ) ‖ ∞ , where θ i = min { β i , k i } , i = 1 , 2, ( u 0 , v 0 ) ∈ W m 1 , p 1 ( V ) × W m 2 , p 2 ( V ) satisfies 0 < ‖ u 0 ‖ ∞ ≤ δ 2 , 0 < ‖ v 0 ‖ ∞ ≤ δ 2 for any x ∈ V , M 5 = 1 k 1 ( M 3 + M 4 ) , M 6 = k 1 − 1 k 1 + 1 k 2 M 4 , h i , ∞ = min x ∈ V h i ( x ) , h i ∞ = max x ∈ V h i ( x ) , i = 1 , 2, μ ∞ = min x ∈ V μ ( x ) ,

‖ ( u λ , v λ ) ‖ = ‖ u λ ‖ m 1 , p 1 + ‖ v λ ‖ m 2 , p 2 , ‖ ( u λ , v λ ) ‖ ∞ = max x ∈ V ∣ ( u λ ( x ) , v λ ( x ) ) ∣ ,

(1.6) Λ 1 = 2 1 − max { p 1 , p 2 } max { p 1 , p 2 } M 5 μ ∞ k 1 − p 1 p 1 h 1 , ∞ k 1 p 1 + M 6 μ ∞ k 2 − p 2 p 2 h 2 , ∞ k 2 p 2 − 1 ,

(1.7) Λ 2 = 2 1 − max { p 1 , p 2 } max { p 1 , p 2 } M 5 μ ∞ k 1 − p 1 p 1 h 1 , ∞ k 1 p 1 + M 6 μ ∞ k 2 − p 2 p 2 h 2 , ∞ k 2 p 2 − 1 6 h 1 , ∞ 1 p 1 ‖ u 0 ‖ L p 1 ( V ) max { p 1 , p 2 } − min { k 1 , k 2 } ,

(1.8) Λ 3 = 2 1 − max { p 1 , p 2 } max { p 1 , p 2 } M 5 μ ∞ k 1 − p 1 p 1 h 1 , ∞ k 1 p 1 + M 6 μ ∞ k 2 − p 2 p 2 h 2 , ∞ k 2 p 2 − 1 3 h 2 , ∞ 1 p 2 ‖ v 0 ‖ L p 2 ( V ) max { p 1 , p 2 } − min { k 1 , k 2 } ,

(1.9) Λ 4 = 1 p 1 max { 1 , h 1 ∞ } ( ‖ ∇ m 1 u 0 ‖ L p 1 ( V ) p 1 + ‖ u 0 ‖ L p 1 ( V ) p 1 ) + 1 p 2 max { 1 , h 2 ∞ } ( ‖ ∇ m 2 v 0 ‖ L p 2 ( V ) p 2 + ‖ v 0 ‖ L p 2 ( V ) p 2 ) M 1 ‖ u 0 ‖ L p 1 ( V ) q 1 ∑ x ∈ V μ ( x ) 1 − q 1 p 1 + M 2 ‖ v 0 ‖ L p 2 ( V ) q 2 ∑ x ∈ V μ ( x ) 1 − q 2 p 2 ,

(1.10) Λ 5 = max 2 2 p 1 + 1 p 1 θ 1 max { C 1 , * , C 2 , * } δ p 1 ( θ 1 − p 1 ) μ ∞ h 1 , ∞ − 2 2 p 1 + 1 p 1 θ 1 max { C 1 , * , C 2 , * } 1 max p 1 q 1 − p 1 , p 2 q 2 − p 2 , × 2 2 p 2 + 1 p 2 θ 2 max { C 1 , * , C 2 , * } δ p 2 ( θ 2 − p 2 ) μ ∞ h 2 , ∞ − 2 2 p 2 + 1 p 2 θ 2 max { C 1 , * , C 2 , * } 1 max p 1 q 1 − p 1 , p 2 q 2 − p 2 ,

(1.11) C 1 , * = ( q 1 − p 1 ) ∑ x ∈ V μ ( x ) p 1 ( max { 1 , h 1 ∞ } ) q 1 q 1 q 1 M 1 p 1 1 q 1 − p 1 ( ‖ ∇ m 1 u 0 ‖ L p 1 ( V ) p 1 + ‖ u 0 ‖ L p 1 ( V ) p 1 ) 1 p 1 ‖ u 0 ‖ L p 1 ( V ) p 1 q 1 q 1 − p 1 ,

(1.12) C 2 , * = ( q 2 − p 2 ) ∑ x ∈ V μ ( x ) p 2 ( max { 1 , h 2 ∞ } ) q 2 q 2 q 2 M 2 p 2 1 q 2 − p 2 ( ‖ ∇ m 2 v 0 ‖ L p 2 ( V ) p 2 + ‖ v 0 ‖ L p 2 ( V ) p 2 ) 1 p 2 ‖ v 0 ‖ L p 2 ( V ) p 2 q 2 q 2 − p 2 .

Remark 1.2

It follows from ( H 1 ) and ( H 3 ) that

  1. F ( x , 0 , 0 ) = 0 for all x ∈ V .

Then by ( H 1 ) , ( H 2 ) , and ( H 4 ) , it is easy to check the following condition:

  1. M 1 ∣ t ∣ q 1 + M 2 ∣ s ∣ q 2 ≤ F ( x , t , s ) ≤ M 5 ∣ t ∣ k 1 + M 6 ∣ s ∣ k 2

    for all x ∈ V and ∣ ( t , s ) ∣ ≤ δ , where M 5 = 1 k 1 ( M 3 + M 4 ) , M 6 = ( k 1 − 1 k 1 + 1 k 2 ) M 4 .

(II) The sub-linear case:

Theorem 1.3

Let G = ( V , E ) be a finite graph, m i ≥ 1 , p i > 1 , i = 1 , 2, λ is a parameter with λ > 0 , h i ( x ) ∈ C ( V , R + ) , i = 1 , 2, F : V × R × R → R satisfies ( H 0 ) , ( H 4 ) , and the following conditions:

  1. There exist q i ∈ ( 1 , p i ) ( i = 1 , 2 ) satisfy min { p 1 , p 2 } > max { q 1 , q 2 } and K i > 0 ( i = 1 , 2 ) such that

    F ( x , t , s ) ≥ K 1 ∣ t ∣ q 1 + K 2 ∣ s ∣ q 2

    for all x ∈ V and ∣ ( t , s ) ∣ ≤ δ ;

  2. There exist k i ∈ ( 1 , q i ) ( i = 1 , 2 ) and K j > 0 ( j = 3 , 4 ) such that

    ∣ F t ( x , t , s ) ∣ ≤ K 3 ∣ t ∣ k 1 − 1 + K 4 ∣ s ∣ k 2 ( k 1 − 1 ) k 1 , ∣ F s ( x , t , s ) ∣ ≤ K 3 ∣ t ∣ k 1 ( k 2 − 1 ) k 2 + K 4 ∣ s ∣ k 2 − 1 ,

    for all x ∈ V and ∣ ( t , s ) ∣ ≤ δ ;

  3. F ( x , t , s ) = F ( x , − t , − s ) for all x ∈ V and ∣ ( t , s ) ∣ ≤ δ .

Then for every λ > 0 , system (1.4) has a sequence of weak solutions { ( u k λ , v k λ ) } with ‖ ( u k λ , v k λ ) ‖ → 0 as k → ∞ .

Remark 1.4

By ( R 1 ) , ( R 2 ) , and ( H 4 ) , it is easy to check that the following condition:

  1. K 1 ∣ t ∣ q 1 + K 2 ∣ s ∣ q 2 ≤ F ( x , t , s ) ≤ K 5 ∣ t ∣ k 1 + K 6 ∣ s ∣ k 2

    for all x ∈ V , ( t , s ) ∈ R 2 with ∣ ( t , s ) ∣ ≤ δ , where K 5 = 1 k 1 ( K 3 + K 4 ) , K 6 = ( k 1 − 1 k 1 + 1 k 2 ) K 4 .

We organize this article as follows. In Section 2, we mainly recall some knowledge for poly-Laplacian on graphs and Sobolev spaces. In Section 3, we complete the proof of Theorem 1.1. In Section 4, we complete the proof of Theorem 1.3. In Section 5, we apply Theorem 1.1 to an example and compute the value of lower bound λ * . Moreover, we also present one example to illustrate Theorem 1.3.

2 Preliminaries

In this section, we recall some notions and important properties about poly-Laplacian on graphs and Sobolev spaces and some useful lemmas. One can see details in [34,36,37].

First, we recall the basic notions of weighted graphs. Let G = ( V , E ) be a finite graph, where V denotes the vertex set and E denotes the edge set. We say that a graph G = ( V , E ) is finite if and only if V and E are both finite sets. Fix an edge weight function ω : E → ( 0 , + ∞ ) satisfying ω x y = ω y x , for any edge { x , y } ∈ E . Let deg ( x ) = Σ x ∼ y ω x y be the degree of x ∈ V , where we write x ∼ y if { x , y } ∈ E . Let μ : V → ( 0 , + ∞ ) be a finite measure on V .

Define C ( V ) as the set of all real functions on V . Then C ( V ) is a finitely dimensional linear space with the usual functions additions and scalar multiplications. For any φ ∈ C ( V ) , we denote

(2.1) ∫ V φ d μ = ∑ x ∈ V μ ( x ) φ ( x ) .

First, for φ 1 , φ 2 ∈ C ( V ) , the associated gradient form reads

(2.2) Γ ( φ 1 , φ 2 ) ( x ) = 1 2 μ ( x ) ∑ y ∼ x ω x y ( φ 1 ( y ) − φ 1 ( x ) ) ( φ 2 ( y ) − φ 2 ( x ) ) .

Sometimes we use the notation ∇ φ 1 ∇ φ 2 = Γ ( φ 1 , φ 2 ) . For any φ ∈ C ( V ) , we write Γ ( φ ) = Γ ( φ , φ ) . Then the length of its gradient is defined as follows:

(2.3) ∣ ∇ φ ∣ = Γ ( φ ) = 1 2 μ ( x ) ∑ y ∼ x ω x y ( φ ( y ) − φ ( x ) ) 2 1 2 .

We have the following equality:

∫ V ( Δ φ ) ϕ d μ = − ∫ V Γ ( φ , ϕ ) d μ , ∀ ϕ ∈ C c ( V ) ,

where the general discrete graph Laplacian Δ : C ( V ) → C ( V ) is defined as follows:

Δ φ ( x ) = 1 μ ( x ) ∑ y ∼ x ω x y ( φ ( y ) − φ ( x ) ) .

By Lemma 2.1 in [36], we know that Δ φ is well defined. Next, the length of m -order gradient of φ ∈ C ( V ) is defined by

∣ ∇ m φ ∣ = ∣ Δ m 2 φ ∣ , when m is even , ∣ ∇ Δ m − 1 2 φ ∣ , when m is odd ,

where m ≥ 1 , ∣ Δ m 2 φ ∣ is defined as the usual absolute of Δ m 2 φ and ∣ ∇ Δ m − 1 2 φ ∣ denotes (2.3) with φ replaced by ∇ Δ m − 1 2 φ . For any p > 1 , the p -Laplacian operator of φ ∈ C ( V ) , denoted by Δ p φ , is defined as the following form:

∫ V ( Δ p φ ) ϕ d μ = − ∫ V ∣ ∇ φ ∣ p − 2 Γ ( φ , ϕ ) d μ , ∀ ϕ ∈ C c ( V ) ,

where Γ ( φ , ϕ ) is defined as in (2.2), C c ( V ) denotes the set of all real functions with compact support, and the integration is defined by (2.1). Δ p : C ( V ) → C ( V ) can also be written as follows:

(2.4) Δ p φ ( x ) = 1 2 μ ( x ) ∑ y ∼ x ( ∣ ∇ φ ∣ p − 2 ( y ) + ∣ ∇ φ ∣ p − 2 ( x ) ) ω x y ( φ ( y ) − φ ( x ) ) .

By Lemma 2.1 in [37], we know that the definition of p -Laplacian operator in (2.4) is reasonable. Now, the generalized graph poly-Laplacian £ m , p : C ( V ) → C ( V ) [34] is introduced as follows:

∫ V ( £ m , p φ ) ϕ d μ = ∫ V ∣ ∇ m φ ∣ p − 2 Γ ( Δ m − 1 2 φ , Δ m − 1 2 ϕ ) d μ , when m is odd , ∫ V ∣ ∇ m φ ∣ p − 2 Δ m 2 φ Δ m 2 ϕ d μ , when m is even ,

where m ≥ 1 , p > 1 , φ ∈ C ( V ) , ϕ ∈ C c ( V ) . It is easy to see that £ m , 2 = ( − Δ ) m and £ 1 , p = − Δ p .

For any 1 ≤ q < ∞ , assume that the completion space of C c ( V ) is L q ( V ) under the norm

‖ φ ‖ L q ( V ) = ∫ V ∣ φ ∣ q d μ 1 q .

Moreover, the completion space of C c ( V ) is W m , p ( V ) under the norm

(2.5) ‖ φ ‖ m , p = ∫ V ( ∣ ∇ m φ ∣ p + h ( x ) ∣ φ ∣ p ) d μ 1 ⁄ p ,

where m ≥ 1 , p > 1 , h ( x ) > 0 for all x ∈ V . Observing that W m , p ( V ) is a finite dimensional linear space and W m , p ( V ) with norm (2.5) is a Banach space. For more details about poly-Laplacian operator and Sobolev spaces on graphs, we refer readers to [34] and the reference therein.

The following inequalities and embedding proposition involving Sobolev spaces will be used frequently in our proofs.

Remark 2.1

It is easy to obtain that

h ∞ ‖ φ ‖ L p ( V ) p ≤ ‖ φ ‖ m , p p ≤ max { 1 , h ∞ } ( ‖ ∇ m φ ‖ L p ( V ) p + ‖ φ ‖ L p ( V ) p ) ,

where h ∞ = min x ∈ V h ( x ) and h ∞ = max x ∈ V h ( x ) .

Lemma 2.2

[31] Let p > 1 . For all φ ∈ W m , p ( V ) , it holds

‖ φ ‖ ∞ ≤ 1 μ ∞ 1 p h ∞ 1 p ‖ φ ‖ m , p ,

where μ ∞ = min x ∈ V μ ( x ) .

Lemma 2.3

[31] Let G = ( V , E ) be a finite graph. Let m be any positive integer and p > 1 . Then the embedding W m , p ( V ) ↪ L q ( V ) is continuous for all 1 ≤ q < + ∞ and

‖ φ ‖ L q ( V ) ≤ ∑ x ∈ V μ ( x ) 1 ⁄ q μ ∞ 1 p h ∞ 1 p ‖ φ ‖ m , p

for all φ ∈ W m , p ( V ) . Moreover, W m , p ( V ) is pre-compact.

Next, we give an inequality, which will be used in the proof of Theorem 1.1.

Lemma 2.4

Let λ > 0 , a > 0 , and b > 0 . Then

λ − a + λ − b ≤ 2 ( 1 + λ − max { a , b } ) .

Proof

If 0 < λ < 1 , we have λ − a + λ − b ≤ 2 λ − max { a , b } . If λ ≥ 1 , it is obvious that λ − a + λ − b ≤ 2 . Then for any λ > 0 , the conclusion holds.□

Now, we recall a special version of the Mountain pass lemma. Let X be a real Banach space, I ∈ C 1 ( X , R ) and c ∈ R . A sequence { φ n } ⊂ X is said to be a ( C ) c -sequence of I in X if

I ( φ n ) → c , ‖ I ′ ( φ n ) ‖ X * ( 1 + ‖ φ n ‖ ) → 0 .

I is said to satisfy the ( C ) c -condition if any ( C ) c -sequence has a convergent subsequence. To prove Theorem 1.1, we need the following Mountain pass lemma.

Lemma 2.5

[38] Let X be a real Banach space, I ∈ C 1 ( X , R ) , w ∈ X , and r > 0 be such that ‖ w ‖ > r and

inf ‖ φ ‖ = r I ( φ ) > I ( 0 ) = 0 ≥ I ( w ) .

Then there exists a ( C ) c -sequence with

c ≔ inf γ ∈ Γ max t ∈ [ 0 , 1 ] I ( γ ( t ) ) , Γ ≔ { γ ∈ C ( [ 0 , 1 ] , X ) : γ ( 0 ) = 0 , γ ( 1 ) = w } .

Next, we recall an extension of Clark’s theorem. Let X be a real Banach space and I ∈ C 1 ( X , R ) . For any sequence { φ n } ⊂ X , if { I ( φ n ) } is bounded and I ′ ( φ n ) → 0 as n → ∞ , then { φ n } is said to be a Palais-Smale sequence of I in X . If any Palais-Smale sequence { φ n } admits a convergent subsequence, then we call that I satisfies Palais-Smale ((PS) for short) condition. To prove Theorem 1.3, we need the following Clark’s theorem.

Lemma 2.6

[39] Let X be a real Banach space and I ∈ C 1 ( X , R ) . Suppose that I satisfies the (PS)-condition, which is even, bounded from below, and I ( 0 ) = 0 . If for any k ∈ N , there exists a k-dimensional subspace Z k of X and ρ k > 0 such that sup Z k ∩ S ρ k I < 0 , where S ρ k = { φ ∈ X ∣ ‖ φ ‖ = ρ k } , then at least one of the following conclusions holds:

  1. There exist a sequence of critical points { φ k } satisfying I ( φ k ) < 0 for all k and ‖ φ k ‖ → 0 as k → ∞ ;

  2. There exist a constant ζ > 0 such that for any 0 < a < ζ there exists a critical point φ such that ‖ φ ‖ = a and I ( φ ) = 0 .

Remark 2.7

[39] Lemma 2.6 implies that there exist a sequence of critical points φ k ≠ 0 such that I ( φ k ) ≤ 0 , I ( φ k ) → 0 , and ‖ φ k ‖ → 0 as k → ∞ .

3 Proofs for the super-linear case

We define the space W ≔ W m 1 , p 1 ( V ) × W m 2 , p 2 ( V ) with the norm ‖ ( u , v ) ‖ = ‖ u ‖ m 1 , p 1 + ‖ v ‖ m 2 , p 2 , where m i ≥ 1 , p i > 1 , i = 1 , 2. Then ( W , ‖ ⋅ ‖ ) is a Banach space. On W , we define the variational functional corresponding to system (1.4) by

I λ ( u , v ) = 1 p 1 ∫ V ( ∣ ∇ m 1 u ∣ p 1 + h 1 ( x ) ∣ u ∣ p 1 ) d μ + 1 p 2 ∫ V ( ∣ ∇ m 2 v ∣ p 2 + h 2 ( x ) ∣ v ∣ p 2 ) d μ − λ ∫ V F ( x , u , v ) d μ .

Since conditions ( H 1 ) and ( H 2 ) imply the behavior of F just near the origin, the functional ∫ V F ( x , u , v ) d μ is not well defined in the Sobolev space W . To deal with this problem, we need to extend F to a proper function F ¯ by the cut-off technique developed by [13]. To adapt system (1.4), we make an extension to R 2 for the cut-off function in [13].

Define τ ( t , s ) ∈ C 1 ( R 2 , [ 0 , 1 ] ) as a cut-off function fulfilling t τ t ′ ( t , s ) ≤ 0 and s τ s ′ ( t , s ) ≤ 0 for all ( t , s ) ∈ R 2 with δ 2 < ∣ ( t , s ) ∣ ≤ δ and

τ ( t , s ) = 1 , if ∣ ( t , s ) ∣ ≤ δ ⁄ 2 , 0 , if ∣ ( t , s ) ∣ > δ .

Define F ¯ : V × R × R → R by

F ¯ ( x , t , s ) = τ ( t , s ) F ( x , t , s ) + ( 1 − τ ( t , s ) ) ( M 5 ∣ t ∣ k 1 + M 6 ∣ s ∣ k 2 ) .

By the definition of τ ( t , s ) , we could obtain the following lemma:

Lemma 3.1

Assume that ( H 0 ) , ( H 2 ) , ( H 3 ) , and ( H 5 ) hold. Then

  1. F ¯ ( x , t , s ) is continuously differentiable in ( t , s ) ∈ R × R for all x ∈ V ;

  2. there exist M 7 > 0 and M 8 > 0 such that

    ∣ F ¯ t ( x , t , s ) ∣ ≤ M 7 ∣ t ∣ k 1 − 1 + M 8 ∣ s ∣ k 2 ( k 1 − 1 ) k 1 , ∣ F ¯ s ( x , t , s ) ∣ ≤ M 7 ∣ t ∣ k 1 ( k 2 − 1 ) k 2 + M 8 ∣ s ∣ k 2 − 1 ,

    for all x ∈ V and ( t , s ) ∈ R 2 ;

  3. 0 < F ¯ ( x , t , s ) ≤ 1 θ 1 t F ¯ t ( x , t , s ) + 1 θ 2 s F ¯ s ( x , t , s )

    for all x ∈ V and ( t , s ) ∈ R 2 \ { ( 0 , 0 ) } , where θ i = min { β i , k i } , i = 1 , 2;

  4. M 1 ∣ t ∣ q 1 + M 2 ∣ s ∣ q 2 ≤ F ¯ ( x , t , s ) ≤ M 5 ∣ t ∣ k 1 + M 6 ∣ s ∣ k 2

    for all x ∈ V and ∣ ( t , s ) ∣ ≤ δ ,

    0 ≤ F ¯ ( x , t , s ) ≤ M 5 ∣ t ∣ k 1 + M 6 ∣ s ∣ k 2

    for all x ∈ V and ( t , s ) ∈ R 2 .

Consider the modified system of (1.4) given by

(3.1) £ m 1 , p 1 u + h 1 ( x ) ∣ u ∣ p 1 − 2 u = λ F ¯ u ( x , u , v ) , x ∈ V , £ m 2 , p 2 v + h 2 ( x ) ∣ v ∣ p 2 − 2 v = λ F ¯ v ( x , u , v ) , x ∈ V .

Define the corresponding functional I ¯ λ : W → R by

I ¯ λ ( u , v ) ≔ 1 p 1 ∫ V ( ∣ ∇ m 1 u ∣ p 1 + h 1 ( x ) ∣ u ∣ p 1 ) d μ + 1 p 2 ∫ V ( ∣ ∇ m 2 v ∣ p 2 + h 2 ( x ) ∣ v ∣ p 2 ) d μ − λ ∫ V F ¯ ( x , u , v ) d μ

for all ( u , v ) ∈ W . By ( H 0 ) ′ , ( H 2 ) ′ , ( H 5 ) ′ , and the continuous embeddings

W m 1 , p 1 ( V ) ↪ L q 1 ( V ) , W m 2 , p 2 ( V ) ↪ L q 2 ( V ) ,

standard arguments show that I ¯ λ is well defined and of class C 1 on W , and

⟨ I ¯ λ ′ ( u , v ) , ( u ˜ , v ˜ ) ⟩ = ∫ V ( £ m 1 , p 1 u u ˜ + h 1 ( x ) ∣ u ∣ p 1 − 2 u u ˜ ) d μ + ∫ V ( £ m 2 , p 2 v v ˜ + h 2 ( x ) ∣ v ∣ p 2 − 2 v v ˜ ) d μ − λ ∫ V F ¯ u ( x , u , v ) u ˜ d μ − λ ∫ V F ¯ v ( x , u , v ) v ˜ d μ

for all ( u , v ) , ( u ˜ , v ˜ ) ∈ W [40]. Hence,

⟨ I ¯ λ ′ ( u , v ) , ( u , v ) ⟩ = ‖ u ‖ m 1 , p 1 p 1 + ‖ v ‖ m 2 , p 2 p 2 − λ ∫ V F ¯ u ( x , u , v ) u d μ − λ ∫ V F ¯ v ( x , u , v ) v d μ

for all ( u , v ) ∈ W . Note that the critical points of I ¯ λ with L ∞ -norm less than or equal to δ ⁄ 2 are also critical points of I λ . Then the weak solutions of system (3.1) with L ∞ -norm less than or equal to δ ⁄ 2 are also weak solutions of problem (1.4).

Lemma 3.2

Assume that ( H 5 ) ′ holds. Then for each λ > 0 , there exists two positive constants ν λ and η λ such that I ¯ λ ( u , v ) ≥ η λ for all ‖ ( u , v ) ‖ = ν λ .

Proof

For any given λ > 0 , ( u , v ) ∈ W with ‖ ( u , v ) ‖ ≤ 1 , by ( H 5 ) ′ , Lemma 2.2, Remark 2.1 and min { k 1 , k 2 } > max { p 1 , p 2 } , we have

(3.2) I ¯ λ ( u , v ) = 1 p 1 ‖ u ‖ m 1 , p 1 p 1 + 1 p 2 ‖ v ‖ m 2 , p 2 p 2 − λ ∫ V F ¯ ( x , u , v ) d μ ≥ 1 max { p 1 , p 2 } ( ‖ u ‖ m 1 , p 1 max { p 1 , p 2 } + ‖ v ‖ m 2 , p 2 max { p 1 , p 2 } ) − λ M 5 ‖ u ‖ ∞ k 1 − p 1 ‖ u ‖ L p 1 ( V ) p 1 − λ M 6 ‖ v ‖ ∞ k 2 − p 2 ‖ v ‖ L p 2 ( V ) p 2 ≥ 2 1 − max { p 1 , p 2 } max { p 1 , p 2 } ‖ ( u , v ) ‖ max { p 1 , p 2 } − λ M 5 ( μ ∞ h 1 , ∞ ) k 1 − p 1 p 1 ‖ u ‖ m 1 , p 1 k 1 − p 1 ‖ u ‖ L p 1 ( V ) p 1 − λ M 6 ( μ ∞ , h 2 , ∞ ) k 2 − p 2 p 2 ‖ v ‖ m 2 , p 2 k 2 − p 2 ‖ v ‖ L p 2 ( V ) p 2 ≥ 2 1 − max { p 1 , p 2 } max { p 1 , p 2 } ‖ ( u , v ) ‖ max { p 1 , p 2 } − λ M 5 μ ∞ k 1 − p 1 p 1 h 1 , ∞ k 1 p 1 ‖ u ‖ m 1 , p 1 k 1 − λ M 6 μ ∞ k 2 − p 2 p 2 h 2 , ∞ k 2 p 2 ‖ v ‖ m 2 , p 2 k 2 ≥ 2 1 − max { p 1 , p 2 } max { p 1 , p 2 } ‖ ( u , v ) ‖ max { p 1 , p 2 } − λ M 5 μ ∞ k 1 − p 1 p 1 h 1 , ∞ k 1 p 1 + M 6 μ ∞ k 2 − p 2 p 2 h 2 , ∞ k 2 p 2 ‖ ( u , v ) ‖ min { k 1 , k 2 } .

Let

ν 0 , λ = 2 1 − max { p 1 , p 2 } λ max { p 1 , p 2 } M 5 μ ∞ k 1 − p 1 p 1 h 1 , ∞ k 1 p 1 + M 6 μ ∞ k 2 − p 2 p 2 h 2 , ∞ k 2 p 2 1 min { k 1 , k 2 } − max { p 1 , p 2 } .

Then by (3.2), ν λ satisfies 0 < ν λ < min { 1 , ν 0 , λ } such that

I ¯ λ ( u , v ) ≥ 2 1 − max { p 1 , p 2 } max { p 1 , p 2 } ν λ max { p 1 , p 2 } − λ M 5 μ ∞ k 1 − p 1 p 1 h 1 , ∞ k 1 p 1 + M 6 μ ∞ k 2 − p 2 p 2 h 2 , ∞ k 2 p 2 ν λ min { k 1 , k 2 } ≔ η λ > 2 1 − max { p 1 , p 2 } max { p 1 , p 2 } ν 0 , λ max { p 1 , p 2 } − λ M 5 μ ∞ k 1 − p 1 p 1 h 1 , ∞ k 1 p 1 + M 6 μ ∞ k 2 − p 2 p 2 h 2 , ∞ k 2 p 2 ν 0 , λ min { k 1 , k 2 } = 0

for all ‖ ( u , v ) ‖ = ν λ .□

Lemma 3.3

Assume that ( H 5 ) ′ holds. Then for each λ > max { Λ 1 , Λ 2 , Λ 3 , Λ 4 } , where Λ 1 , Λ 2 , Λ 3 , and Λ 4 are defined in (1.6), (1.7), (1.8), and (1.9), respectively, there exists ( u 0 , v 0 ) ∈ W m 1 , p 1 ( V ) \ { 0 } × W m 2 , p 2 ( V ) \ { 0 } with u 0 > 0 , v 0 > 0 , and ‖ ( u 0 , v 0 ) ‖ ∞ ≤ δ such that ‖ ( u 0 , v 0 ) ‖ > ν λ and I ¯ λ ( u 0 , v 0 ) < 0 .

Proof

Let ( u 0 , v 0 ) ∈ W m 1 , p 1 ( V ) \ { 0 } × W m 2 , p 2 ( V ) \ { 0 } with u 0 > 0 , v 0 > 0 for any x ∈ V and satisfies ‖ u 0 ‖ ∞ ≤ δ 2 and ‖ v 0 ‖ ∞ ≤ δ 2 . It is easy to see that ν 0 , λ < 1 for any λ > Λ 1 . We could take ν λ = 1 2 ν 0 , λ . By Remark 2.1, we have ‖ u 0 ‖ m 1 , p 1 > 1 3 ν λ for all λ > Λ 2 and ‖ v 0 ‖ m 2 , p 2 > 2 3 ν λ for all λ > Λ 3 , respectively. Then ‖ ( u 0 , v 0 ) ‖ = ‖ u 0 ‖ m 1 , p 1 + ‖ v 0 ‖ m 2 , p 2 > 1 3 ν λ + 2 3 ν λ = ν λ for all λ > max { Λ 2 , Λ 3 } . It follows from Remark 2.1, ( H 5 ) ′ and Hölder’s inequality that

I ¯ λ ( u 0 , v 0 ) = 1 p 1 ‖ u 0 ‖ m 1 , p 1 p 1 + 1 p 2 ‖ v 0 ‖ m 2 , p 2 p 2 − λ ∫ V F ¯ ( x , u 0 , v 0 ) d μ ≤ 1 p 1 max { 1 , h 1 ∞ } ( ‖ ∇ m 1 u 0 ‖ L p 1 ( V ) p 1 + ‖ u 0 ‖ L p 1 ( V ) p 1 ) + 1 p 2 max { 1 , h 2 ∞ } ( ‖ ∇ m 2 v 0 ‖ L p 2 ( V ) p 2 + ‖ v 0 ‖ L p 2 ( V ) p 2 ) − λ M 1 ∑ x ∈ V μ ( x ) 1 − q 1 p 1 ‖ u 0 ‖ L p 1 ( V ) q 1 + M 2 ∑ x ∈ V μ ( x ) 1 − q 2 p 2 ‖ v 0 ‖ L p 2 ( V ) q 2 < 0

for all λ > Λ 4 .□

It follows from Lemmas 3.2, 3.3, and the fact I ¯ λ ( 0 , 0 ) = 0 that I ¯ λ has a mountain pass geometry, that is, setting

Γ ≔ { γ ∈ C ( [ 0 , 1 ] , W ) : γ ( 0 ) = ( 0 , 0 ) , γ ( 1 ) = ( u 0 , v 0 ) } .

Obviously, Γ ≠ ∅ . By Lemma 2.5, for the mountain pass level

c λ ≔ inf γ ∈ Γ max t ∈ [ 0 , 1 ] I ¯ λ ( γ ( t ) ) ,

there exists a ( C ) c λ -sequence { ( u n , v n ) } ≔ { ( u n , λ , v n , λ ) } of I ¯ λ in W , that is,

(3.3) I ¯ λ ( u n , v n ) → c λ and ( 1 + ‖ ( u n , v n ) ‖ ) ‖ I ¯ λ ′ ( u n , v n ) ‖ W * → 0 , as n → ∞ .

Moreover, Lemma 3.2 implies that c λ > 0 .

Lemma 3.4

Assume that ( H 3 ) ′ holds. Then the ( C ) c λ -sequence { ( u n , v n ) } has a convergent subsequence for any given λ > 0 .

Proof

By (3.3), we have

‖ ( u n , v n ) ‖ ‖ I ¯ λ ′ ( u n , v n ) ‖ W * → 0 as n → ∞ ,

which implies

(3.4) I ¯ λ ′ ( u n , v n ) , 1 θ 1 u n , 1 θ 2 v n ≤ 1 min { θ 1 , θ 2 } ‖ I ¯ λ ′ ( u n , v n ) ‖ W * ‖ ( u n , v n ) ‖ → 0 as n → ∞ .

Then by (3.3), (3.4), and ( H 3 ) ′ , for n large enough and any λ > 0 , we have

(3.5) c λ + 1 ≥ I ¯ λ ( u n , v n ) − I ¯ λ ′ ( u n , v n ) , 1 θ 1 u n , 1 θ 2 v n = 1 p 1 − 1 θ 1 ‖ u n ‖ m 1 , p 1 p 1 + 1 p 2 − 1 θ 2 ‖ v n ‖ m 1 , p 2 p 2 + λ ∫ V 1 θ 1 u n F ¯ u ( x , u n , v n ) + 1 θ 2 v n F ¯ v ( x , u n , v n ) − F ¯ ( x , u n , v n ) d μ ≥ 1 p 1 − 1 θ 1 ‖ u n ‖ m 1 , p 1 p 1 + 1 p 2 − 1 θ 2 ‖ v n ‖ m 1 , p 2 p 2 ,

which implies that { u n } and { v n } are bounded in W m 1 , p 1 ( V ) and W m 2 , p 2 ( V ) , respectively. As both W m 1 , p 1 ( V ) and W m 2 , p 2 ( V ) are precompact, up to a subsequence of { ( u n , v n ) } , still denoted by { ( u n , v n ) } , there exists ( u λ , v λ ) ∈ W such that u n → u λ in W m 1 , p 1 ( V ) and v n → v λ in W m 2 , p 2 ( V ) , respectively.□

Lemma 3.4 shows that ( u n , v n ) → ( u λ , v λ ) in W , which together with (3.3) and the continuity of I ¯ λ and I ¯ λ ′ imply that I ¯ λ ( u λ , v λ ) = c λ and I ¯ λ ′ ( u λ , v λ ) = 0 . Moreover, by I ¯ λ ( 0 , 0 ) = 0 and c λ > 0 , we have ( u λ , v λ ) ≠ ( 0 , 0 ) . Thus, ( u λ , v λ ) is a nontrivial critical point of I ¯ λ with the critical value c λ . Next, we will show that ( u λ , v λ ) precisely is the nontrivial weak solution of system (1.4).

Lemma 3.5

Assume that ( H 3 ) ′ and ( H 5 ) ′ hold. Then there exist C 1 , * > 0 and C 2 , * > 0 independent of λ > 0 such that

‖ u λ ‖ ∞ ≤ p 1 θ 1 max { C 1 , * , C 2 , * } ( θ 1 − p 1 ) μ ∞ h 1 , ∞ 1 p 1 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 1 , ‖ v λ ‖ ∞ ≤ p 2 θ 2 max { C 1 , * , C 2 , * } ( θ 2 − p 2 ) μ ∞ h 2 , ∞ 1 p 2 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 2 ,

where C 1 , * and C 2 , * are defined in (1.11), and (1.12), respectively.

Proof

Let s ∈ [ 0 , 1 ] . Then ‖ s ( u 0 , v 0 ) ‖ ∞ ≤ ‖ ( u 0 , v 0 ) ‖ ∞ ≤ δ , where ( u 0 , v 0 ) is given in Lemma 3.3. For all λ > 0 , by ( H 5 ) ′ and Hölder’s inequality, we have

(3.6) I ¯ λ ( s u 0 , s v 0 ) = 1 p 1 ‖ u 0 ‖ m 1 , p 1 p 1 s p 1 + 1 p 2 ‖ v 0 ‖ m 2 , p 2 p 2 s p 2 − λ ∫ V F ¯ ( x , s u 0 , s v 0 ) d μ ≤ 1 p 1 ‖ u 0 ‖ m 1 , p 1 p 1 s p 1 + 1 p 2 ‖ v 0 ‖ m 2 , p 2 p 2 s p 2 − λ M 1 s q 1 ∫ V ∣ u 0 ∣ q 1 d μ − λ M 2 s q 2 ∫ V ∣ v 0 ∣ q 2 d μ ≤ 1 p 1 ‖ u 0 ‖ m 1 , p 1 p 1 s p 1 − λ M 1 ∑ x ∈ V μ ( x ) 1 − q 1 p 1 ‖ u 0 ‖ L p 1 ( V ) q 1 s q 1 + 1 p 2 ‖ v 0 ‖ m 2 , p 2 p 2 s p 2 − λ M 2 ∑ x ∈ V μ ( x ) 1 − q 2 p 2 ‖ v 0 ‖ L p 2 ( V ) q 2 s q 2 .

Define g i : [ 0 , + ∞ ) → R , i = 1 , 2, by

g 1 ( s ) = 1 p 1 ‖ u 0 ‖ m 1 , p 1 p 1 s p 1 − λ M 1 ∑ x ∈ V μ ( x ) 1 − q 1 p 1 ‖ u 0 ‖ L p 1 ( V ) q 1 s q 1 , g 2 ( s ) = 1 p 2 ‖ v 0 ‖ m 2 , p 2 p 2 s p 2 − λ M 2 ∑ x ∈ V μ ( x ) 1 − q 2 p 2 ‖ v 0 ‖ L p 2 ( V ) q 2 s q 2 .

For any λ > 0 , let

g 1 ′ ( s ) = s p 1 − 1 ‖ u 0 ‖ m 1 , p 1 p 1 − λ q 1 M 1 ∑ x ∈ V μ ( x ) 1 − q 1 p 1 ‖ u 0 ‖ L p 1 ( V ) q 1 s q 1 − p 1 = 0 .

Then we have s 1 λ = ∑ x ∈ V μ ( x ) 1 p 1 ‖ u 0 ‖ m 1 , p 1 p 1 λ q 1 M 1 ‖ u 0 ‖ L p 1 ( V ) q 1 1 q 1 − p 1 > 0 . It follows from Remark 2.1 that

(3.7) max s ≥ 0 g 1 ( s ) = g 1 ( s 1 λ ) = ( q 1 − p 1 ) ∑ x ∈ V μ ( x ) p 1 ‖ u 0 ‖ m 1 , p 1 ‖ u 0 ‖ L p 1 ( V ) p 1 q 1 q 1 − p 1 ( q 1 q 1 M 1 p 1 ) 1 p 1 − q 1 λ − p 1 q 1 − p 1

(3.7) ≤ ( q 1 − p 1 ) ∑ x ∈ V μ ( x ) p 1 ( ‖ ∇ m 1 u 0 ‖ L p 1 ( V ) p 1 + ‖ u 0 ‖ L p 1 ( V ) p 1 ) 1 p 1 ‖ u 0 ‖ L p 1 ( V ) p 1 q 1 q 1 − p 1 × ( max { 1 , h 1 ∞ } ) q 1 q 1 q 1 M 1 p 1 1 q 1 − p 1 λ − p 1 q 1 − p 1 = C 1 , * λ − p 1 q 1 − p 1 .

Similarly, we have

(3.8) max s ≥ 0 g 2 ( s ) = g 2 ( s 2 λ ) ≤ C 2 , * λ − p 2 q 2 − p 2 ,

where s 2 λ = ∑ x ∈ V μ ( x ) 1 p 2 ‖ v 0 ‖ m 2 , p 2 p 2 λ q 2 M 2 ‖ v 0 ‖ L p 2 ( V ) q 2 1 q 2 − p 2 > 0 . By the definition of c λ , (3.6), (3.7), and (3.8), we have

(3.9) c λ ≤ max s ∈ [ 0 , 1 ] I ¯ λ ( s u 0 , s v 0 ) ≤ max s ≥ 0 g 1 ( s ) + max s ≥ 0 g 2 ( s ) ≤ max { C 1 , * , C 2 , * } λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 .

Note that I ¯ λ ′ ( u λ , v λ ) , ( 1 θ 1 u λ , 1 θ 2 v λ ) = 0 . Similar to the argument in (3.5), we have

‖ u λ ‖ m 1 , p 1 ≤ p 1 θ 1 θ 1 − p 1 1 p 1 c λ 1 p 1 , ‖ v λ ‖ m 2 , p 2 ≤ p 2 θ 2 θ 2 − p 2 1 p 2 c λ 1 p 2 ,

which combining with (3.9) connote that

(3.10) ‖ u λ ‖ m 1 , p 1 ≤ p 1 θ 1 max { C 1 , * , C 2 , * } θ 1 − p 1 1 p 1 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 1

and

(3.11) ‖ v λ ‖ m 2 , p 2 ≤ p 2 θ 2 max { C 1 , * , C 2 , * } θ 2 − p 2 1 p 2 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 2 .

By Lemma 2.2, we further obtain the following results:

(3.12) ‖ u λ ‖ ∞ ≤ p 1 θ 1 max { C 1 , * , C 2 , * } ( θ 1 − p 1 ) μ ∞ h 1 , ∞ 1 p 1 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 1

and

(3.13) ‖ v λ ‖ ∞ ≤ p 2 θ 2 max { C 1 , * , C 2 , * } ( θ 2 − p 2 ) μ ∞ h 2 , ∞ 1 p 2 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 2 .

The proof is completed.□

Proof of Theorem 1.1

It follows from the fact q i > p i > 1 , i = 1 , 2 and Lemma 2.4 that

λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 ≤ 2 1 + λ − max p 1 q 1 − p 1 , p 2 q 2 − p 2 ,

which together with Lemma 3.5 implies that ‖ ( u λ , v λ ) ‖ ∞ ≤ ‖ u λ ‖ ∞ + ‖ v λ ‖ ∞ ≤ δ 4 + δ 4 = δ 2 for each λ > Λ 5 , where Λ 5 is defined in (1.10). Therefore, ( u λ , v λ ) ∈ W is a nontrivial weak solution of system (1.4). Furthermore, (3.10)–(3.13) imply that

‖ ( u λ , v λ ) ‖ ≤ p 1 θ 1 max { C 1 , * , C 2 , * } θ 1 − p 1 1 p 1 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 1 + p 2 θ 2 max { C 1 , * , C 2 , * } θ 2 − p 2 1 p 2 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 2

and

‖ ( u λ , v λ ) ‖ ∞ ≤ p 1 θ 1 max { C 1 , * , C 2 , * } ( θ 1 − p 1 ) μ ∞ h 1 , ∞ 1 p 1 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 1 + p 2 θ 2 max { C 1 , * , C 2 , * } ( θ 2 − p 2 ) μ ∞ h 2 , ∞ 1 p 2 λ − p 1 q 1 − p 1 + λ − p 2 q 2 − p 2 1 p 2 .

Thus, we have lim λ → ∞ ‖ ( u λ , v λ ) ‖ = 0 = lim λ → ∞ ‖ ( u λ , v λ ) ‖ ∞ . □

4 Proofs for the sublinear case

Define ρ ( t , s ) ∈ C 1 ( R 2 , [ 0 , 1 ] ) as a cut-off function satisfying

ρ ( t , s ) = 1 , if ∣ ( t , s ) ∣ ≤ δ ⁄ 2 , 0 , if ∣ ( t , s ) ∣ > δ .

Define F ˜ : V × R × R → R by

F ˜ ( x , t , s ) = ρ ( t , s ) F ( x , t , s ) + ( 1 − ρ ( t , s ) ) ( K 1 ∣ t ∣ q 1 + K 2 ∣ s ∣ q 2 ) .

By ( H 0 ) , ( R 2 ) – ( R 4 ) , and the definition of ρ ( t , s ) , we could obtain the following lemma:

Lemma 4.1

Assume that ( H 0 ) , ( R 2 ) – ( R 4 ) hold. Then

  1. F ˜ ( x , t , s ) is continuously differentiable in R × R for all x ∈ V ;

  2. there exist K 7 > 0 and K 8 > 0 such that

    ∣ F ˜ t ( x , t , s ) ∣ ≤ K 7 ( ∣ t ∣ q 1 − 1 + ∣ t ∣ k 1 − 1 ) + K 8 ∣ s ∣ k 2 ( k 1 − 1 ) k 1 , ∣ F ˜ s ( x , t , s ) ∣ ≤ K 7 ∣ t ∣ k 1 ( k 2 − 1 ) k 2 + K 8 ( ∣ s ∣ q 2 − 1 + ∣ s ∣ k 2 − 1 ) ,

    for all x ∈ V and ( t , s ) ∈ R 2 ;

  3. F ˜ ( x , t , s ) = F ˜ ( x , − t , − s ) for all x ∈ V and ( t , s ) ∈ R 2 ;

  4. K 1 ∣ t ∣ q 1 + K 2 ∣ s ∣ q 2 ≤ F ˜ ( x , t , s ) ≤ max { K 1 , K 5 } ( ∣ t ∣ q 1 + ∣ t ∣ k 1 ) + max { K 2 , K 6 } ( ∣ s ∣ q 2 + ∣ s ∣ k 2 )

    for all x ∈ V and ( t , s ) ∈ R 2 .

Consider the modified system of (1.4) given by

(4.1) £ m 1 , p 1 u + h 1 ( x ) ∣ u ∣ p 1 − 2 u = λ F ˜ u ( x , u , v ) , x ∈ V , £ m 2 , p 2 v + h 2 ( x ) ∣ v ∣ p 2 − 2 v = λ F ˜ v ( x , u , v ) , x ∈ V .

Define the corresponding functional I ˜ λ : W → R by

I ˜ λ ( u , v ) ≔ 1 p 1 ∫ V ( ∣ ∇ m 1 u ∣ p 1 + h 1 ( x ) ∣ u ∣ p 1 ) d μ + 1 p 2 ∫ V ( ∣ ∇ m 2 v ∣ p 2 + h 2 ( x ) ∣ v ∣ p 2 ) d μ − λ ∫ V F ˜ ( x , u , v ) d μ

for all ( u , v ) ∈ W . By ( R 0 ) ′ – ( R 4 ) ′ and the continuous embeddings

W m i , p i ( V ) ↪ L q i ( V ) , W m i , p i ( V ) ↪ L k i ( V ) , i = 1 , 2 ,

standard arguments show that I ˜ λ is well defined, even and of class C 1 on W , and

⟨ I ˜ λ ′ ( u , v ) , ( ϕ 1 , ϕ 2 ) ⟩ = ∫ V ( £ m 1 , p 1 u ϕ 1 + h 1 ( x ) ∣ u ∣ p 1 − 2 u ϕ 1 ) d μ + ∫ V ( £ m 2 , p 2 v ϕ 2 + h 2 ( x ) ∣ v ∣ p 2 − 2 v ϕ 2 ) d μ − λ ∫ V F ˜ u ( x , u , v ) ϕ 1 d μ − λ ∫ V F ˜ v ( x , u , v ) ϕ 2 d μ

for all ( u , v ) , ( ϕ 1 , ϕ 2 ) ∈ W [40]. Hence,

⟨ I ˜ λ ′ ( u , v ) , ( u , v ) ⟩ = ‖ u ‖ m 1 , p 1 p 1 + ‖ v ‖ m 2 , p 2 p 2 − λ ∫ V F ˜ u ( x , u , v ) u d μ − λ ∫ V F ˜ v ( x , u , v ) v d μ

for all ( u , v ) ∈ W . Note that the critical points of I ˜ λ with L ∞ -norm less than or equal to δ ⁄ 2 are also critical points of I λ . Then the weak solutions of system (4.1) with L ∞ -norm less than or equal to δ ⁄ 2 are also weak solutions of problem (1.4).

Proof of Theorem 1.3

Obviously, I ˜ λ ∈ C 1 ( W , R ) is even and I ˜ λ ( 0 , 0 ) = 0 . For any fixed λ > 0 , using ( R 4 ) ′ , we obtain

I ˜ λ ( u , v ) ≥ 1 p 1 ‖ u ‖ m 1 , p 1 p 1 − λ max { K 1 , K 5 } ∫ V ( ∣ u ∣ q 1 + ∣ u ∣ k 1 ) d μ + 1 p 2 ‖ v ‖ m 2 , p 2 p 2 − λ max { K 2 , K 6 } ∫ V ( ∣ v ∣ q 2 + ∣ v ∣ k 2 ) d μ ,

which estimate by Lemma 2.3 leads to

I ˜ λ ( u , v ) ≥ 1 p 1 ‖ u ‖ m 1 , p 1 p 1 − λ max { K 1 , K 5 } ∑ x ∈ V μ ( x ) 1 μ ∞ q 1 p 1 h 1 , ∞ q 1 p 1 ‖ u ‖ m 1 , p 1 q 1 + 1 μ ∞ k 1 p 1 h 1 , ∞ k 1 p 1 ‖ u ‖ m 1 , p 1 k 1 + 1 p 2 ‖ v ‖ m 2 , p 2 p 2 − λ max { K 2 , K 6 } ∑ x ∈ V μ ( x ) 1 μ ∞ q 2 p 2 h 2 , ∞ q 2 p 2 ‖ v ‖ m 2 , p 2 q 2 + 1 μ ∞ k 2 p 2 h 2 , ∞ k 2 p 2 ‖ v ‖ m 2 , p 2 k 2 ,

which together with the fact p i > q i > k i > 1 ( i = 1 , 2 ) implies that I ˜ λ ( u , v ) → + ∞ as ‖ ( u , v ) ‖ → ∞ . So, I ˜ λ is coercive and bounded form below on W . Suppose that { ( u n , v n ) } is a Palais-Smale sequence of I ˜ λ on W , that is,

{ I ˜ λ ( u n , v n ) } is bounded , I ˜ λ ′ ( u n , v n ) → 0 as n → ∞ .

Then the coercivity of I ˜ λ implies that { ( u n , v n ) } is bounded on W . Due to both W m 1 , p 1 ( V ) and W m 2 , p 2 ( V ) are pre-compact, up to a subsequence of { ( u n , v n ) } , still denoted by { ( u n , v n ) } , there exists ( u ˜ λ , v ˜ λ ) ∈ W such that u n → u ˜ λ in W m 1 , p 1 ( V ) and v n → v ˜ λ in W m 2 , p 2 ( V ) , respectively. Hence, functional I ˜ λ satisfies the Palais-Smale condition for all λ > 0 . Let Z k be a k -dimensional subspace of W for any k ∈ N . Then all norms are equivalent in Z k . Hence, there exist positive constants c 1 and c 2 such that

(4.2) ‖ u ‖ L q 1 ( V ) q 1 ≥ c 1 ‖ u ‖ m 1 , p 1 q 1 , ‖ v ‖ L q 2 ( V ) q 2 ≥ c 2 ‖ v ‖ m 2 , p 2 q 2

for all ( u , v ) ∈ Z k . For any given 0 < ρ k < min { 1 , ρ λ } ,

ρ λ = p 1 p 2 p 1 + p 2 λ min { K 1 c 1 , K 2 c 2 } 2 1 − max { q 1 , q 2 } 1 min { p 1 , p 2 } − max { q 1 , q 2 } ,

define

S ρ k = { ( u , v ) ∈ W ∣ ‖ ( u , v ) ‖ = ρ k } .

By ( R 4 ) ′ , (4.2), and the fact min { p 1 , p 2 } > max { q 1 , q 2 } , we have

I ˜ λ ( u , v ) = 1 p 1 ‖ u ‖ m 1 , p 1 p 1 + 1 p 2 ‖ v ‖ m 2 , p 2 p 2 − λ ∫ V F ˜ ( x , u , v ) d μ ≤ p 1 + p 2 p 1 p 2 ‖ ( u , v ) ‖ min { p 1 , p 2 } − λ K 1 ∫ V ∣ u ∣ q 1 d μ − λ K 2 ∫ V ∣ v ∣ q 2 d μ

≤ p 1 + p 2 p 1 p 2 ‖ ( u , v ) ‖ min { p 1 , p 2 } − λ K 1 c 1 ‖ u ‖ m 1 , p 1 q 1 − λ K 2 c 2 ‖ v ‖ m 2 , p 2 q 2 ≤ p 1 + p 2 p 1 p 2 ‖ ( u , v ) ‖ min { p 1 , p 2 } − λ K 1 c 1 ‖ u ‖ m 1 , p 1 max { q 1 , q 2 } − λ K 2 c 2 ‖ v ‖ m 2 , p 2 max { q 1 , q 2 } ≤ p 1 + p 2 p 1 p 2 ‖ ( u , v ) ‖ min { p 1 , p 2 } − λ min { K 1 c 1 , K 2 c 2 } ( ‖ u ‖ m 1 , p 1 max { q 1 , q 2 } + ‖ v ‖ m 2 , p 2 max { q 1 , q 2 } ) ≤ p 1 + p 2 p 1 p 2 ‖ ( u , v ) ‖ min { p 1 , p 2 } − λ min { K 1 c 1 , K 2 c 2 } 2 1 − max { q 1 , q 2 } ‖ ( u , v ) ‖ max { q 1 , q 2 } < 0

for any ( u , v ) ∈ Z k ∩ S ρ k and λ > 0 . So sup Z k ∩ S ρ k I ˜ λ < 0 . Hence, Lemma 2.6 and Remark 2.7 show that system (4.1) has infinitely many nonzero solutions { ( u k λ , v k λ ) } such that ‖ ( u k λ , v k λ ) ‖ → 0 as k → ∞ . By Lemma 2.2, we further obtain

‖ ( u k λ , v k λ ) ‖ ∞ ≤ ‖ u k λ ‖ ∞ + ‖ v k λ ‖ ∞ ≤ max 1 μ ∞ 1 p 1 h 1 , ∞ 1 p 1 , 1 μ ∞ 1 p 2 h 2 , ∞ 1 p 2 ‖ ( u k λ , v k λ ) ‖ → 0 as k → ∞ .

So there exists a sufficiently large integer k 0 > 0 such that ‖ ( u k λ , v k λ ) ‖ ∞ ≤ δ 2 for all k ≥ k 0 , which implies that { ( u k λ , v k λ ) } k 0 ∞ is a sequence of weak solutions of the original system (1.4) for each fixed λ > 0 .□

5 Examples

In this section, we present two examples that satisfy the conditions of Theorems 1.1 and 1.3, respectively.

Example 5.1

Let G = ( V , E ) is a finite graph, specifically,

V = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 } , E = { { x 1 , x 2 } , { x 1 , x 3 } , { x 1 , x 4 } , { x 1 , x 5 } , { x 2 , x 3 } , { x 2 , x 6 } , { x 3 , x 4 } , { x 4 , x 7 } , { x 5 , x 6 } , { x 5 , x 7 } } , μ ( x 1 ) = 2 , μ ( x 2 ) = 1 , μ ( x 3 ) = μ ( x 4 ) = 3 , μ ( x 5 ) = μ ( x 6 ) = 5 , μ ( x 7 ) = 6 and deg ( x 1 ) = 6 .

Then ∫ V 1 d μ = Σ x ∈ V μ ( x ) = 25 . Let A = { x ∈ V ∣ x ∼ x 1 } and ♯ A = 4 , where ♯ A is the number of elements in the set A . Let

e 1 ( x ) = e 2 ( x ) = 1 , if x = x 1 , 0 , if x ≠ x 1 .

Choose

(5.1) u 0 = δ μ ∞ 1 2 h 1 , ∞ 1 2 2 17 ( max { 1 , h 1 ∞ } ) 1 2 e 1 and v 0 = δ μ ∞ 1 2 h 2 , ∞ 1 2 2 17 ( max { 1 , h 2 ∞ } ) 1 2 e 2 .

Let δ = 1 , m 1 = m 2 = 1 , p 1 = p 2 = 2 , and h 1 ( x ) = h 2 ( x ) = 1 for all x ∈ V . Consider the following system:

(5.2) − Δ u + u = λ F u ( x , u , v ) , x ∈ V , − Δ v + v = λ F v ( x , u , v ) , x ∈ V ,

where

F ( x , t , s ) = σ ( t , s ) ( ∣ t ∣ 6 + ∣ s ∣ 6 ) + ( 1 − σ ( t , s ) ) ( ∣ t ∣ 4 + ∣ s ∣ 4 )

for all ( x , t , s ) ∈ V × R × R with σ ( t , s ) defined by

σ ( t , s ) = 1 , if ∣ ( t , s ) ∣ ≤ 1 , sin π ( t 2 + s 2 − 16 ) 2 450 , if 1 < ∣ ( t , s ) ∣ ≤ 4 , 0 , if ∣ ( t , s ) ∣ > 4 .

So,

F ( x , t , s ) = ∣ t ∣ 6 + ∣ s ∣ 6 , if ∣ ( t , s ) ∣ ≤ 1 , ( ∣ t ∣ 6 + ∣ s ∣ 6 ) sin π ( t 2 + s 2 − 16 ) 2 450 + 1 − sin π ( t 2 + s 2 − 16 ) 2 450 ( ∣ t ∣ 4 + ∣ s ∣ 4 ) , if 1 < ∣ ( t , s ) ∣ ≤ 4 , ∣ t ∣ 4 + ∣ s ∣ 4 , if ∣ ( t , s ) ∣ > 4 .

By Theorem 1.1, we can obtain that system (5.2) has at least a nontrivial weak solution ( u λ , v λ ) for each λ > Λ 4 ≈ 89523333.3 and lim λ → ∞ ‖ ( u λ , v λ ) ‖ = 0 = lim λ → ∞ ‖ ( u λ , v λ ) ‖ ∞ .

In fact, we can verify that F satisfies the conditions in Theorem 1.1. Let

A ( t , s ) ≔ 6 ∣ t ∣ 4 t sin π ( t 2 + s 2 − 16 ) 2 450 + 4 1 − sin π ( t 2 + s 2 − 16 ) 2 450 b ( x ) ∣ t ∣ 2 t + ( ∣ t ∣ 6 + ∣ s ∣ 6 ) 2 π t ( t 2 + s 2 − 16 ) 225 cos π ( t 2 + s 2 − 16 ) 2 450 − ( ∣ t ∣ 4 + ∣ s ∣ 4 ) 2 π t ( t 2 + s 2 − 16 ) 225 cos π ( t 2 + s 2 − 16 ) 2 450 ,

and

B ( t , s ) ≔ 6 ∣ s ∣ 4 s sin π ( t 2 + s 2 − 16 ) 2 450 + 4 1 − sin π ( t 2 + s 2 − 16 ) 2 450 ∣ s ∣ 2 s + ( ∣ t ∣ 6 + ∣ s ∣ 6 ) 2 π s ( t 2 + s 2 − 16 ) 225 cos π ( t 2 + s 2 − 16 ) 2 450 − ( ∣ t ∣ 4 + ∣ s ∣ 4 ) 2 π s ( t 2 + s 2 − 16 ) 225 cos π ( t 2 + s 2 − 16 ) 2 450 .

It is easy to see that F satisfies ( H 0 ) and

F t ( x , t , s ) = 6 ∣ t ∣ 4 t , if ∣ ( t , s ) ∣ ≤ 1 , A ( t , s ) , if 1 < ∣ ( t , s ) ∣ ≤ 4 , 4 ∣ t ∣ 2 t , if ∣ ( t , s ) ∣ > 4 ,

F s ( x , t , s ) = 6 ∣ s ∣ 4 s , if ∣ ( t , s ) ∣ ≤ 1 , B ( t , s ) , if 1 < ∣ ( t , s ) ∣ ≤ 4 , 4 ∣ s ∣ 2 s , if ∣ ( t , s ) ∣ > 4 .

Hence, we obtain

F ( x , t , s ) = ∣ t ∣ 6 + ∣ s ∣ 6 ≥ ∣ t ∣ 7 + ∣ s ∣ 7 , for all x ∈ V and ∣ ( t , s ) ∣ ≤ 1 .

Then ( H 1 ) holds with q 1 = q 2 = 7 > 2 = p 1 = p 2 , M 1 = M 2 = 1 . We also have

∣ F t ( x , t , s ) ∣ = 6 ∣ t ∣ 5 ≤ 6 ( ∣ t ∣ 4 + ∣ s ∣ 4 ) , for all x ∈ V and ∣ ( t , s ) ∣ ≤ 1 .

Similarly,

∣ F s ( x , t , s ) ∣ ≤ 6 ( ∣ t ∣ 4 + ∣ s ∣ 4 ) , for all x ∈ V and ∣ ( t , s ) ∣ ≤ 1 .

Then ( H 2 ) holds with k 1 = k 2 = 5 ∈ ( 2, 7 ) and M 3 = M 4 = 6 . In particular, taking β 1 = β 2 = 3 > 2 = p 1 = p 2 , we have

F ( x , t , s ) = ∣ t ∣ 6 + ∣ s ∣ 6 ≤ 2 ( ∣ t ∣ 6 + ∣ s ∣ 6 ) = 1 β 1 t F t ( x , t , s ) + 1 β 2 s F s ( x , t , s )

for all x ∈ R N and ∣ ( t , s ) ∣ ≤ 1 . Thus, assumption ( H 3 ) is satisfied.

Next, we complete the value of λ ∗ by the formulas in Theorem 1.1. It follows that

(5.3) ‖ ∇ e 1 ‖ L 2 ( V ) 2 = ∑ x ∈ V ∣ ∇ e 1 ∣ 2 ( x ) μ ( x ) = ∑ x ∈ V 1 2 μ ( x ) ∑ y ∼ x ω x y ( e 1 ( y ) − e 1 ( x ) ) 2 μ ( x ) = 1 2 μ ( x 1 ) ∑ y ∼ x 1 ω x 1 y μ ( x 1 ) + ∑ x ∼ x 1 1 2 μ ( x ) ∑ x 1 ∼ x ω x x 1 μ ( x ) = 1 2 deg ( x 1 ) ( 1 + ♯ A ) = 15 = ‖ ∇ e 2 ‖ L 2 ( V ) 2 .

Similarly, we have

(5.4) ‖ e 1 ‖ L 2 ( V ) 2 = ‖ e 2 ‖ L 2 ( V ) 2 = μ ( x 1 ) = 2 .

By Lemma 2.2, Remark 2.1, (5.3), (5.4), and (5.1), we have

0 < ‖ u 0 ‖ ∞ ≤ δ 2 and 0 < ‖ v 0 ‖ ∞ ≤ δ 2 .

Moreover,

‖ ∇ u 0 ‖ L 2 ( V ) 2 = 15 δ 2 μ ∞ h 1 , ∞ 68 max { 1 , h 1 ∞ } , ‖ ∇ v 0 ‖ L 2 ( V ) 2 = 15 δ 2 μ ∞ h 2 , ∞ 68 max { 1 , h 2 ∞ } , ‖ u 0 ‖ L 2 ( V ) 2 = δ 2 μ ∞ h 1 , ∞ 34 max { 1 , h 1 ∞ } , ‖ v 0 ‖ L 2 ( V ) 2 = δ 2 μ ∞ h 2 , ∞ 34 max { 1 , h 2 ∞ } .

Then by θ 1 = θ 2 = 3 , M 5 = 12 5 , M 6 = 6 , h 1 , ∞ = h 2 , ∞ = 1 = h 1 ∞ = h 2 ∞ , μ ∞ = min x ∈ V μ ( x ) = 1 , (1.11) and (1.12), we have

C 1 , * = C 2 , * = 125 2 17 14 7 5 ,

and by (1.6)–(1.10),

Λ 1 = 5 168 , Λ 2 = 5 168 ( 6 ) − 3 ( 34 ) 3 2 , Λ 3 = 5 168 ( 3 ) − 3 ( 34 ) 3 2 , Λ 4 = 17 136 ( 25 ) 5 2 ( 34 ) 7 2 , Λ 5 = ( 2 ) 9 ( 3 ) 5 2 ( 5 ) 15 2 ( 17 ) 7 2 ( 7 ) 7 5 − 3 ( 5 ) 3 ( 2 ) 18 5 ( 17 ) 7 5 5 2 .

Compared Λ 1 , Λ 2 , Λ 3 , Λ 4 , and Λ 5 , it is easy to see λ ∗ = Λ 4 ≈ 89523333.3 .

Remark 5.2

Through the conclusion of Theorem 1.1 and the calculations for the parameter λ in Example 5.1, it is easy to see the following interesting phenomenon: the specific lower bound of the parameter on a finite graph is related to the structure of the graph, such as the degree and the measure of any point in V . However, in the Euclidean setting, in general, the specific lower bound is only related to the measure of domain [17]. The fundamental reason for this distinction lies in the difference between the definitions of integrals on the graph and in Euclidean space.

Example 5.3

Let G = ( V , E ) is a finite graph, δ = 1 , m 1 = m 2 = 1 , p 1 = 2 , p 2 = 3 , and h 1 ( x ) = h 2 ( x ) = 1 for all x ∈ V . Consider the following system:

(5.5) − Δ u + u = λ F u ( x , u , v ) , x ∈ V , − Δ 3 v + ∣ v ∣ v = λ F v ( x , u , v ) , x ∈ V ,

where

F ( x , t , s ) = 3 4 ( ∣ t ∣ 4 3 + ∣ s ∣ 4 3 )

for all ( x , t , s ) ∈ V × R × R . By Theorem 1.2, we can obtain that system (5.5) has a sequence of weak solutions { ( u k λ , v k λ ) } with ‖ ( u k λ , v k λ ) ‖ → 0 as k → ∞ for every λ > 0 .

In fact, we can verify that F satisfies the conditions in Theorem 1.3. It is easy to see that ( H 0 ) , ( H 4 ) , and ( R 3 ) hold. We also have ( R 1 ) holds with q 1 = q 2 = 5 3 ∈ ( 1 , 2 ) and K 1 = K 2 = 3 4 . It follows that

∣ F t ( x , t , s ) ∣ = ∣ t ∣ 1 3 ≤ 2 ( ∣ t ∣ 1 4 + ∣ s ∣ 1 4 ) , for all x ∈ V and ∣ ( t , s ) ∣ ≤ 1 .

Similarly,

∣ F s ( x , t , s ) ∣ ≤ 2 ( ∣ t ∣ 1 4 + ∣ s ∣ 1 4 ) , for all x ∈ V and ∣ ( t , s ) ∣ ≤ 1 .

Then ( R 2 ) holds with k 1 = k 2 = 5 4 ∈ ( 1 , 5 3 ) and K 3 = K 4 = 2 .

Acknowledgments

The authors sincerely thank the reviewers for their valuable comments.

  1. Funding information: This project was supported by Yunnan Fundamental Research Projects (grant No: 202301AT070465) and Xingdian Talent Support Program for Young Talents of Yunnan Province.

  2. Author contributions: Wanting Qi and Xingyong Zhang contributed to the main manuscript equally.

  3. Conflict of interest: The authors declare that they have no conflict of interest.

  4. Ethical approval: Not applicable.

  5. Data availability statement: No data were used in this study.

References

[1] L. Boccardo and D. G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 9 (2002), 309–323, DOI: https://doi.org/10.1007/978-3-319-02856-9_33. 10.1007/s00030-002-8130-0Search in Google Scholar

[2] A. Djellit and S. Tas, Existence of solutions for a class of elliptic systems in RN involving the p-Laplacian, Electron. J. Differential Equations 2003 (2003), no. 56, 1–8. Search in Google Scholar

[3] A. Djellit and S. Tas, On some nonlinear elliptic systems, Nonlinear Anal. 59 (2004), no. 5, 695–706, DOI: https://doi.org/10.1016/j.na.2004.07.029. 10.1016/S0362-546X(04)00279-2Search in Google Scholar

[4] A. Djellit and S. Tas, Quasilinear elliptic systems with critical Sobolev exponents in RN, Nonlinear Anal. 66 (2007), no. 7, 1485–1497, DOI: https://doi.org/10.1016/j.na.2006.02.005. 10.1016/j.na.2006.02.005Search in Google Scholar

[5] N. S. Papageorgiou, V. D. Rădulescu, and W. Zhang, Global existence and multiplicity for nonlinear Robin eigenvalue problems, Results Math. 78 (2023), 133, DOI: https://doi.org/10.1007/s00025-023-01912-8. 10.1007/s00025-023-01912-8Search in Google Scholar

[6] N. S. Papageorgiou, J. Zhang, and W. Zhang, Global existence and multiplicity of solutions for nonlinear singular eigenvalue problems, Discrete Contin. Dyn. Syst. Ser. S, 18 (2025), no. 6, 1440–1456, DOI: https://doi.org/10.3934/dcdss.2024018. 10.3934/dcdss.2024018Search in Google Scholar

[7] J. Zhang and Y. Zhang, An infinite sequence of localized semiclassical states for nonlinear Maxwell-Dirac system, J. Geom. Anal. 34 (2024), 277, DOI: https://doi.org/10.1007/s12220-024-01724-4. 10.1007/s12220-024-01724-4Search in Google Scholar

[8] J. I. Díaz, Nonlinear partial differential equations and free boundaries, Pitman Research Notes in Mathematics 1 (1985), 106. Search in Google Scholar

[9] P. Drábek, The p-Laplacian-mascot of nonlinear analysis, Acta Math. Univ. Comenian. 76 (2007), 85–98. Search in Google Scholar

[10] G. B. Li and H. S. Zhou, Multiple solutions to p-Laplacian problems with asymptotic nonlinearity as up−1 at infinity, J. Lond. Math. Soc. 65 (2002), no. 1, 123–138, DOI: https://doi.org/10.1112/S0024610701002708. 10.1112/S0024610701002708Search in Google Scholar

[11] G. Bonanno and G. M. Bisci, Infinitely many solutions for a Dirichlet problem involving the p-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 737–752, DOI: https://doi.org/10.1017/S0308210509000845. 10.1017/S0308210509000845Search in Google Scholar

[12] M. Ramaswamy and R. Shivaji, Multiple positive solutions for classes of p-Laplacian equations, Differential Integral Equations 17 (2004), no. 11–12, 1255–1261, DOI: https://doi.org/10.57262/die/1356060244. 10.57262/die/1356060244Search in Google Scholar

[13] D. G. Costa and Z. Q. Wang, Multiplicity results for a class of superlinear elliptic problems, Proc. Amer. Math. Soc. 133 (2005), no. 3, 787–794, DOI: https://doi.org/10.1090/S0002-9939-04-07635-X. 10.1090/S0002-9939-04-07635-XSearch in Google Scholar

[14] C. Huang and G. Jia, Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Math. Anal. Appl. 472 (2019), no. 1, 705–727, DOI: https://doi.org/10.1016/j.jmaa.2018.11.048. 10.1016/j.jmaa.2018.11.048Search in Google Scholar

[15] J. M. D. E. Medeiros and U. Severo, On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations, J. Math. Anal. Appl. 342 (2008), no. 1, 432–445, DOI: https://doi.org/10.1016/j.jmaa.2007.11.058. 10.1016/j.jmaa.2007.11.058Search in Google Scholar

[16] E. S. Medeiros and U. B. Severo, On the existence of signed solutions for a quasilinear elliptic problem in RN, Mat. Complement. 32 (2007), 193–205, DOI: https://doi.org/10.21711/231766362007/rmc3210. 10.21711/231766362007/rmc3210Search in Google Scholar

[17] D. Y. Kang, C. L. Liu, and X. Y. Zhang, Existence of solutions for Kirchhoff-type fractional Dirichlet problem with p-Laplacian, Mathematics 8 (2020), no. 1, 106, DOI: https://doi.org/10.3390/math8010106. 10.3390/math8010106Search in Google Scholar

[18] A. E. Chakik, A. Elmoataz, and X. Desquesnes, Mean curvature flow on graphs for image and manifold restoration and enhancement, Signal Processing 105 (2014), 449–463, DOI: http://dx.doi.org/10.1016/j.sigpro.2014.04.029. 10.1016/j.sigpro.2014.04.029Search in Google Scholar

[19] E. B. Curtis, and J. A. Morrow, Determining the resistors in a network, SIAM J. Appl. Math. 50 (1990), no 3, 918–930, DOI: https://doi.org/10.1137/0150055. 10.1137/0150055Search in Google Scholar

[20] X. Desquesnes, A. Elmoataz, and O. Lézoray, Eikonal equation adaptation on weighted graphs: fast geometric diffusion process for local and non-local image and data processing, J. Math. Imaging Vision 46 (2013), no. 2, 238–257, DOI: https://doi.org/10.1007/s10851-012-0380-9. 10.1007/s10851-012-0380-9Search in Google Scholar

[21] A. Elmoataz, X. Desquesnes, and O. Lézoray, Non-local morphological PDEs and p-Laplacian equation on graphs with applications in image processing and machine learning, IEEE J. Sel. Top. Signal Process. 6 (2012), no. 7, 764–779, DOI: https://doi.org/10.1109/JSTSP.2012.2216504. 10.1109/JSTSP.2012.2216504Search in Google Scholar

[22] A. Elmoataz, F. Lozes, and M. Toutain, Nonlocal PDEs on graphs: from Tug-of-War games to unified interpolation on images and point clouds, J. Math. Imaging Vision, 57 (2017), no. 3, 381–401, DOI: https://doi.org/10.1007/s10851-016-0683-3. 10.1007/s10851-016-0683-3Search in Google Scholar

[23] A. Elmoataz, M. Toutain, and D. Tenbrinck, On the p-Laplacian and ∞-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci. 8 (2015), no. 4, 2412–2451, DOI: https://doi.org/10.1137/15M1022793. 10.1137/15M1022793Search in Google Scholar

[24] Y. S. Chung, Y. S. Lee, and S. Y. Chung, Extinction and positivity of the solutions of the heat equations with absorption on networks, J. Math. Anal. Appl. 380 (2011), no. 2, 642–652, DOI: https://doi.org/10.1016/j.jmaa.2011.03.006. 10.1016/j.jmaa.2011.03.006Search in Google Scholar

[25] Y. Lin and Y. T. Wu, The existence and nonexistence of global solutions for a semilinear heat equation on graphs, Calc. Var. Partial Differential Equations 56 (2017), 102, DOI: https://doi.org/10.1007/s00526-017-1204-y. 10.1007/s00526-017-1204-ySearch in Google Scholar

[26] W. J. Liu, K. W. Chen, and J. Yu, Extinction and asymptotic behavior of solutions for the ω-heat equation on graphs with source and interior absorption, J. Math. Anal. Appl. 435 (2016), no. 1, 112–132, DOI: https://doi.org/10.1016/j.jmaa.2015.10.024. 10.1016/j.jmaa.2015.10.024Search in Google Scholar

[27] P. X. Huang, On uniqueness class for a heat equation on graphs, J. Math. Anal. Appl. 393 (2012), no. 2, 377–388, DOI: https://doi.org/10.1016/j.jmaa.2012.04.026. 10.1016/j.jmaa.2012.04.026Search in Google Scholar

[28] Q. Xin, L. Xu, and C. Mu, Blow-up for the ω-heat equation with Dirichlet boundary conditions and a reaction term on graphs, Appl. Anal. 93 (2014), no. 8, 1691–1701, DOI: https://doi.org/10.1080/00036811.2013.842640. 10.1080/00036811.2013.842640Search in Google Scholar

[29] A. Grigor’yan, Y. Lin, and Y. Y. Yang, Kazdan-Warner equation on graph, Calc. Var. Partial Differential Equations 55 (2016), no. 4, DOI: https://doi.org/10.1007/s00526-016-1042-3. 10.1007/s00526-016-1042-3Search in Google Scholar

[30] H. B. Ge and W. F. Jiang, Kazdan-Warner equation on infinite graphs, J. Korean Math. Soc. 55 (2018), no. 5, 1091–1101, DOI: https://doi.org/10.4134/JKMS.j170561. Search in Google Scholar

[31] X. C. Zhang, X. Y. Zhang, J. P. Xie, and X. L. Yu, Existence and multiplicity of nontrivial solutions for poly-Laplacian systems on finite graphs, Bound. Value Probl. 2022 (2022), 32, DOI: https://doi.org/10.1186/s13661-022-01613-1. 10.1186/s13661-022-01613-1Search in Google Scholar

[32] X. L. Yu, X. Y. Zhang, J. P. Xie, and X. C. Zhang, Existence of nontrivial solutions for a class of poly-Laplacian system with mixed nonlinearity on graphs, Math. Methods Appl. Sci. 47 (2024), no. 4, 1750–1763, DOI: https://doi.org/10.1002/mma.9621. 10.1002/mma.9621Search in Google Scholar

[33] P. Yang and X. Y. Zhang, Existence of nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearities on locally finite graphs, Electron. Res. Arch. 31 (2023), no. 12, 7473–7495, DOI: https://doi.org/10.3934/era.2023377. 10.3934/era.2023377Search in Google Scholar

[34] A. Grigor’yan, Y. Lin, and Y. Y. Yang, Yamabe type equations on graphs, J. Differential Equations 261 (2017), no. 9, 4924–4943, DOI: https://doi.org/10.1016/j.jde.2016.07.011. 10.1016/j.jde.2016.07.011Search in Google Scholar

[35] A. Grigor’yan, Y. Lin, and Y. Y. Yang, Existence of positive solutions to some nonlinear equations on locally finite graphs, Sci. China Math. 60 (2017), no. 7, 1311–1324, DOI: https://doi.org/10.1007/s11425-016-0422-y. 10.1007/s11425-016-0422-ySearch in Google Scholar

[36] N. Zhang and L. Zhao, Convergence of ground state solutions for nonlinear Schrödinger equations on graphs, Sci. China-Math. 61 (2018), 1481–1494, DOI: https://doi.org/10.1007/s11425-017-9254-7. 10.1007/s11425-017-9254-7Search in Google Scholar

[37] X. L. Han and M. Q. Shao, p-Laplacian equations on locally finite graphs, Acta Math. Sin. (Engl. Ser.) 37 (2021), no. 11, 1645–1678, DOI: https://doi.org/10.1007/s10114-021-9523-5. 10.1007/s10114-021-9523-5Search in Google Scholar

[38] I. Ekeland, Convexity Methods in Hamilton Mechanics, Springer-Verlag, New York, 1990. 10.1007/978-3-642-74331-3Search in Google Scholar

[39] Z. L. Liu and Z. Q. Wang, On Clarkas theorem and its applications to partially sublinear problems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 32 (2015), no. 5, 1015–1037, DOI: https://doi.org/10.1016/J.ANIHPC.2014.05.002. 10.1016/j.anihpc.2014.05.002Search in Google Scholar

[40] P. Yang and X. Y. Zhang, Existence and multiplicity of nontrivial solutions for a (p,q)-Laplacian system on locally finite graphs, Taiwanese J. Math. 28 (2024), no. 3, 551–588, DOI: https://doi.org/10.11650/tjm/240201. 10.11650/tjm/240201Search in Google Scholar
Received: 2024-09-07
Revised: 2025-02-22
Accepted: 2025-06-17
Published Online: 2025-07-31

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Research Articles
  2. On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
  3. Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
  4. Grand Triebel-Lizorkin-Morrey spaces
  5. Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
  6. Proofs of two conjectures involving sums of normalized Narayana numbers
  7. On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
  8. New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
  9. Some inequalities for rational function with prescribed poles and restricted zeros
  10. Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
  11. Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
  12. Fixed point results for contractions of polynomial type
  13. Estimation for spatial semi-functional partial linear regression model with missing response at random
  14. Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
  15. New forms of bilateral inequalities for K-g-frames
  16. Rate of pole detection using Padé approximants to polynomial expansions
  17. Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
  18. Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
  19. Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
  20. New topologies derived from the old one via operators
  21. Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
  22. Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
  23. Nonparametric expectile shortfall regression for functional data
  24. Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
  25. A generalized p-Laplacian problem with parameters
  26. A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2
  27. Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
  28. ω ℒ -biprojective and ω ¯ -contractible Banach algebras
  29. Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
  30. Solutions of several general quadratic partial differential-difference equations in ℂ2
  31. Inequalities for the generalized trigonometric functions with respect to weighted power mean
  32. Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
  33. Hankel determinants for q-starlike functions connected with q-sine function
  34. System of partial differential hemivariational inequalities involving nonlocal boundary conditions
  35. A new family of multivalent functions defined by certain forms of the quantum integral operator
  36. A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
  37. Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties
  38. Behavior of spatial curves under different transformations in Euclidean 4-space
  39. Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
  40. A new kind of Durrmeyer-Stancu-type operators
  41. A study of generalized Mittag-Leffler-type function of arbitrary order
  42. On the approximation of Kantorovich-type Szàsz-Charlier operators
  43. Split quaternion Fourier transforms for two-dimensional real invariant field
  44. Quantum injectivity of G-frames in Hilbert spaces
  45. Some results on disjointly weakly compact sets
  46. On Motzkin sequence spaces via q-analog and compact operators
  47. Existence and multiplicity of nontrivial solutions for Schrödinger-Bopp-Podolsky systems with critical nonlinearity in ℝ3
  48. Stability analysis of linear time-invariant difference-differential system with constant and distributed delays
  49. The discriminant of quasi m-boundary singularities
  50. Norm constrained empirical portfolio optimization with stochastic dominance: Robust optimization non-asymptotics
  51. Fuzzy stability of multi-additive mappings
  52. On inequalities involving n-polynomial exponential-type convex functions
  53. Singularities of multiplicative spherical Darboux image and multiplicative rectifying developable surface
  54. A golden ratio technique for equilibrium problem in reflexive Banach spaces
  55. A parallel inertial three-step iteration monotone hybrid algorithm for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with graphs applicable to signal recovery problems
  56. Multiple and unique nontrivial solutions for fractional differential equations with singular property and derivatives contained in the nonlinear term
  57. New soliton solutions for a nonlinear complex hyperbolic Schrödinger dynamical equation with a truncated M-fractional derivative
  58. On a generalization of derangement polynomials and numbers
  59. The description of entire solutions of complex PDEs and PDDEs
  60. A modified RMIL conjugate gradient-based projection algorithm for constrained nonlinear equations: application to image denoising
  61. Fast solution strategies for time-space fractional linear complementarity problems governing American options pricing
  62. Existence results for Robin problems involving p(x)-Laplacian-like operators with convection term
  63. On asymptotic behaviors of a specific cubic functional equation and its hyperstability
  64. The description of entire solutions for some class of complex nonlinear partial differential equation (systems) in C 2
  65. Variations in the geometry of the basins of escape in a modified Hénon–Heiles potential
  66. A Rothe method for a viscoelastic contact problem involving time-fractional derivatives in locking materials
  67. Upper and lower solution method for a higher order ϕ-Laplacian BVPs on an infinite interval
  68. Weyl almost periodic functions on time scales and their Fourier series
  69. Integrable system of null curve and Betchov-Da Rios equation
  70. Fekete–Szegö problems for (β, Φ)-spirllike mapping of complex order γ in Banach space
  71. Modulated convergence: a deferred approach
  72. Infinitely many solutions for an instantaneous and non-instantaneous fourth-order differential system with local assumptions
  73. Existence and nonexistence of normalized solutions for the Biharmonic equation with combined nonlinearities
  74. Ekeland’s variational principle for interval-valued functions with an α-level set in Kaleva-Seikkala’s type fuzzy metric spaces
  75. On Kurzweil integral of fuzzy number valued functions with two variables
  76. On split common null point and common fixed point problems for multivalued demicontractive mappings
  77. Approximation by weighted Durrmeyer-type max-product neural network operators
  78. A new predictor-corrector interior-point algorithm for semidefinite optimization
  79. Densities of measures: fine properties and examples
  80. Review Articles
  81. Characterization generalized derivations of tensor products of nonassociative algebras
  82. On the performance of the new minimax shrinkage estimators for a normal mean vector
  83. Special Issue on Differential Equations and Numerical Analysis - Part II
  84. Existence and optimal control of Hilfer fractional evolution equations
  85. Persistence of a unique periodic wave train in convecting shallow water fluid
  86. Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
  87. Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
  88. Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
  89. Stability and bifurcation analysis of a modified chemostat model
  90. Some new quantum derivatives and integrals with their applications in integral error bounds
  91. Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
  92. Analytic solutions of a generalized complex multi-dimensional system with fractional order
  93. Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model
  94. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part II
  95. Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
  96. On the sum form functional equation related to diversity index
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part II
  98. Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
  99. Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
  100. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
  101. Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
  102. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
  103. Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
  104. New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
  105. Special Issue on Approximation Theory and Special Functions 2024 conference
  106. Ulam-type stability for Caputo-type fractional delay differential equations
  107. Faster approximation to multivariate functions by combined Bernstein-Taylor operators
  108. (λ, ψ)-Bernstein-Kantorovich operators
  109. Some special functions and cylindrical diffusion equation on α-time scale
  110. (q, p)-Mixing Bloch maps
  111. Orthogonalizing q-Bernoulli polynomials
  112. On better approximation order for the max-product Meyer-König and Zeller operator
  113. Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
  114. A note on linear compositions of the Mellin convolution operators in the weighted Mellin-Lebesgue spaces
  115. A new perspective on generalized Laguerre polynomials
  116. Global existence of semilinear system of Klein-Gordon equations in anti-de Sitter spacetime
  117. Estimates for Durrmeyer-type exponential sampling series in Mellin-Orlicz spaces
  118. ℐ-αβ-statistical relative uniform convergence for double sequences and its applications
  119. New developments for the Jacobi polynomials
  120. Generalization of Sheffer-λ polynomials
  121. Fractional calculus containing certain bivariate Mittag-Leffler kernel with respect to function
  122. A new type of soft multi rough sets
  123. Special Issue on Variational Methods and Nonlinear PDEs
  124. A note on mean field type equations
  125. Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
  126. Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
  127. Bifurcation, quasi-periodic, and wave solutions to the fractional model of optical fibers in communication systems
  128. Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth
  129. Ground state solutions to singularly perturbed Chern-Simons-Schrödinger systems with a neutral scalar field
  130. Weak solutions to an asymptotic equation of the variational sine-Gordon equation
  131. Multiplicity of positive solutions for a concave-convex fractional elliptic system with critical growth
https://doi.org/10.1515/dema-2025-0148
Keywords for this article
coupled system on finite graphs; poly-Laplacian; local nonlinearity near origin; Mountain pass lemma; Clark’s theorem; existence; infinitely many nontrivial solutions
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
  3. Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
  4. Grand Triebel-Lizorkin-Morrey spaces
  5. Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
  6. Proofs of two conjectures involving sums of normalized Narayana numbers
  7. On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
  8. New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
  9. Some inequalities for rational function with prescribed poles and restricted zeros
  10. Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
  11. Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
  12. Fixed point results for contractions of polynomial type
  13. Estimation for spatial semi-functional partial linear regression model with missing response at random
  14. Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
  15. New forms of bilateral inequalities for K-g-frames
  16. Rate of pole detection using Padé approximants to polynomial expansions
  17. Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
  18. Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
  19. Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
  20. New topologies derived from the old one via operators
  21. Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
  22. Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
  23. Nonparametric expectile shortfall regression for functional data
  24. Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
  25. A generalized p-Laplacian problem with parameters
  26. A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2
  27. Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
  28. ω ℒ -biprojective and ω ¯ -contractible Banach algebras
  29. Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
  30. Solutions of several general quadratic partial differential-difference equations in ℂ2
  31. Inequalities for the generalized trigonometric functions with respect to weighted power mean
  32. Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
  33. Hankel determinants for q-starlike functions connected with q-sine function
  34. System of partial differential hemivariational inequalities involving nonlocal boundary conditions
  35. A new family of multivalent functions defined by certain forms of the quantum integral operator
  36. A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
  37. Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties
  38. Behavior of spatial curves under different transformations in Euclidean 4-space
  39. Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
  40. A new kind of Durrmeyer-Stancu-type operators
  41. A study of generalized Mittag-Leffler-type function of arbitrary order
  42. On the approximation of Kantorovich-type Szàsz-Charlier operators
  43. Split quaternion Fourier transforms for two-dimensional real invariant field
  44. Quantum injectivity of G-frames in Hilbert spaces
  45. Some results on disjointly weakly compact sets
  46. On Motzkin sequence spaces via q-analog and compact operators
  47. Existence and multiplicity of nontrivial solutions for Schrödinger-Bopp-Podolsky systems with critical nonlinearity in ℝ3
  48. Stability analysis of linear time-invariant difference-differential system with constant and distributed delays
  49. The discriminant of quasi m-boundary singularities
  50. Norm constrained empirical portfolio optimization with stochastic dominance: Robust optimization non-asymptotics
  51. Fuzzy stability of multi-additive mappings
  52. On inequalities involving n-polynomial exponential-type convex functions
  53. Singularities of multiplicative spherical Darboux image and multiplicative rectifying developable surface
  54. A golden ratio technique for equilibrium problem in reflexive Banach spaces
  55. A parallel inertial three-step iteration monotone hybrid algorithm for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with graphs applicable to signal recovery problems
  56. Multiple and unique nontrivial solutions for fractional differential equations with singular property and derivatives contained in the nonlinear term
  57. New soliton solutions for a nonlinear complex hyperbolic Schrödinger dynamical equation with a truncated M-fractional derivative
  58. On a generalization of derangement polynomials and numbers
  59. The description of entire solutions of complex PDEs and PDDEs
  60. A modified RMIL conjugate gradient-based projection algorithm for constrained nonlinear equations: application to image denoising
  61. Fast solution strategies for time-space fractional linear complementarity problems governing American options pricing
  62. Existence results for Robin problems involving p(x)-Laplacian-like operators with convection term
  63. On asymptotic behaviors of a specific cubic functional equation and its hyperstability
  64. The description of entire solutions for some class of complex nonlinear partial differential equation (systems) in C 2
  65. Variations in the geometry of the basins of escape in a modified Hénon–Heiles potential
  66. A Rothe method for a viscoelastic contact problem involving time-fractional derivatives in locking materials
  67. Upper and lower solution method for a higher order ϕ-Laplacian BVPs on an infinite interval
  68. Weyl almost periodic functions on time scales and their Fourier series
  69. Integrable system of null curve and Betchov-Da Rios equation
  70. Fekete–Szegö problems for (β, Φ)-spirllike mapping of complex order γ in Banach space
  71. Modulated convergence: a deferred approach
  72. Infinitely many solutions for an instantaneous and non-instantaneous fourth-order differential system with local assumptions
  73. Existence and nonexistence of normalized solutions for the Biharmonic equation with combined nonlinearities
  74. Ekeland’s variational principle for interval-valued functions with an α-level set in Kaleva-Seikkala’s type fuzzy metric spaces
  75. On Kurzweil integral of fuzzy number valued functions with two variables
  76. On split common null point and common fixed point problems for multivalued demicontractive mappings
  77. Approximation by weighted Durrmeyer-type max-product neural network operators
  78. A new predictor-corrector interior-point algorithm for semidefinite optimization
  79. Densities of measures: fine properties and examples
  80. Review Articles
  81. Characterization generalized derivations of tensor products of nonassociative algebras
  82. On the performance of the new minimax shrinkage estimators for a normal mean vector
  83. Special Issue on Differential Equations and Numerical Analysis - Part II
  84. Existence and optimal control of Hilfer fractional evolution equations
  85. Persistence of a unique periodic wave train in convecting shallow water fluid
  86. Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
  87. Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
  88. Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
  89. Stability and bifurcation analysis of a modified chemostat model
  90. Some new quantum derivatives and integrals with their applications in integral error bounds
  91. Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
  92. Analytic solutions of a generalized complex multi-dimensional system with fractional order
  93. Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model
  94. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part II
  95. Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
  96. On the sum form functional equation related to diversity index
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part II
  98. Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
  99. Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
  100. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
  101. Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
  102. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
  103. Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
  104. New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
  105. Special Issue on Approximation Theory and Special Functions 2024 conference
  106. Ulam-type stability for Caputo-type fractional delay differential equations
  107. Faster approximation to multivariate functions by combined Bernstein-Taylor operators
  108. (λ, ψ)-Bernstein-Kantorovich operators
  109. Some special functions and cylindrical diffusion equation on α-time scale
  110. (q, p)-Mixing Bloch maps
  111. Orthogonalizing q-Bernoulli polynomials
  112. On better approximation order for the max-product Meyer-König and Zeller operator
  113. Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
  114. A note on linear compositions of the Mellin convolution operators in the weighted Mellin-Lebesgue spaces
  115. A new perspective on generalized Laguerre polynomials
  116. Global existence of semilinear system of Klein-Gordon equations in anti-de Sitter spacetime
  117. Estimates for Durrmeyer-type exponential sampling series in Mellin-Orlicz spaces
  118. ℐ-αβ-statistical relative uniform convergence for double sequences and its applications
  119. New developments for the Jacobi polynomials
  120. Generalization of Sheffer-λ polynomials
  121. Fractional calculus containing certain bivariate Mittag-Leffler kernel with respect to function
  122. A new type of soft multi rough sets
  123. Special Issue on Variational Methods and Nonlinear PDEs
  124. A note on mean field type equations
  125. Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
  126. Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
  127. Bifurcation, quasi-periodic, and wave solutions to the fractional model of optical fibers in communication systems
  128. Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth
  129. Ground state solutions to singularly perturbed Chern-Simons-Schrödinger systems with a neutral scalar field
  130. Weak solutions to an asymptotic equation of the variational sine-Gordon equation
  131. Multiplicity of positive solutions for a concave-convex fractional elliptic system with critical growth
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Winner of the OpenAthens UX Award 2026
Winner of the OpenAthens UX Award 2026
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 30.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/dema-2025-0148/html
Scroll to top button